[ver] 4 [sty] [files] [charset] 82 ANSI (Windows, IBM CP 1252) [revisions] 0 [prn] Epson LQ-850 [port] LPT1.OS2 [lang] 2 [desc] Chapter VII, second revised version 792321919 42 788508563 18201 21 6164 38095 173 223 1 [fopts] 0 1 0 0 [lnopts] 2 Body Text 1 [docopts] 5 2 [GramStyle] Academic Writing [tag] Body Text 2 [fnt] Times New Roman 240 0 49152 [algn] 1 1 0 0 0 [spc] 33 273 1 0 0 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 0 0 1 1 0 0 0 0 [nfmt] 280 1 2 . , $ Body Text 0 0 [tag] Body Single 3 [fnt] Times New Roman 240 0 49152 [algn] 1 1 0 0 0 [spc] 33 273 1 0 0 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 0 0 1 1 0 0 0 0 [nfmt] 280 1 2 . , $ Body Single 0 0 [tag] Bullet 4 [fnt] Times New Roman 240 0 49152 [algn] 1 1 0 288 288 [spc] 33 273 1 0 0 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 0 <*0> 360 1 1 0 0 0 0 [nfmt] 272 1 2 . , $ Bullet 0 0 [tag] Bullet 1 5 [fnt] Times New Roman 240 0 49152 [algn] 1 1 288 288 288 [spc] 33 273 1 0 0 1 100 [brk] 4 [line] 8 0 1 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33279 0 65535 0 65280 0 65280 178 65280 255 32768 255 0 255 16225 255 129 255 255 255 255 128 65535 255 0 0 1 49407 206 49407 182 49919 129 65535 194 65474 145 65473 213 65473 255 61378 255 61920 255 57568 255 49121 255 49663 253 49407 228 65535 255 36751 143 33535 160 33023 128 49151 24 65535 128 65408 128 65410 202 65408 255 58017 255 57792 255 49087 255 40930 255 32767 255 33535 194 63479 247 32896 128 16639 112 33279 65 33279 0 65535 0 65345 50 65346 199 65280 255 49215 255 40833 226 32896 255 33472 255 17151 249 16639 160 61423 239 29298 114 255 0 17151 30 25314 0 49087 0 65280 0 65280 178 57344 224 32768 255 33089 255 16705 255 17026 255 255 255 255 128 57825 225 24415 95 194 0 8417 0 16289 0 41377 0 49664 0 49408 150 49408 194 25088 225 0 255 65 255 129 255 161 159 192 127 53970 210 20303 79 129 0 160 0 17026 0 33153 0 32768 0 40704 130 33280 128 16384 128 0 128 34 161 65 194 128 128 130 64 49344 192 16448 64 45520 161 41440 117 45266 106 49856 124 49538 104 49281 151 49791 188 45681 207 40863 224 41408 224 40930 222 37359 235 40930 200 45746 178 12079 47 24703 79 25249 82 25216 16 33410 63 25151 31 24892 62 24631 94 16656 96 24674 161 16738 129 12640 129 8544 98 8546 82 41634 162 0 0 0 824 984 428 154 153 155 151 10 0 8 7751 24 7751 32 7751 40 7751 16 7751 48 7751 0 0 .sdw 197 158 0 [frm] 17 537526272 3580 8264 8772 11811 0 1 3 0 0 0 0 0 0 0 0 16777215 8 7 2140 5192 3731 [frmname] Frame15 [frmlay] 11811 5192 1 0 0 1 8264 0 0 2 0 0 0 0 1 3580 8772 0 [isd] .X8 .sdw 1 1 0 0 4979 61984 100 0 384 258 0 0 1 16 7743 24 7743 32 7743 40 7743 48 7743 56 7743 8 7743 9324 0 2505 1648 7686 5194 13 65391 0 5 65280 255 65535 255 0 0 0 0 20 0 0 21504 29549 21024 28269 0 0 0 0 0 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 1 360 180 360 720 1 0 0 11766 7222 7302 0 7159 1882 7159 115 7279 192 7279 418 7263 487 7263 288 7279 1 255 0 17151 30 33279 0 65535 0 65280 0 65280 178 65280 255 32768 255 0 255 16225 255 129 255 255 255 255 128 65535 255 0 0 1 49407 206 49407 182 49919 129 65535 194 65474 145 65473 213 65473 255 61378 255 61920 255 57568 255 49121 255 49663 253 49407 228 65535 255 36751 143 33535 160 33023 128 49151 24 65535 128 65408 128 65410 202 65408 255 58017 255 57792 255 49087 255 40930 255 32767 255 33535 194 63479 247 32896 128 16639 112 33279 65 33279 0 65535 0 65345 50 65346 199 65280 255 49215 255 40833 226 32896 255 33472 255 17151 249 16639 160 61423 239 29298 114 255 0 17151 30 25314 0 49087 0 65280 0 65280 178 57344 224 32768 255 33089 255 16705 255 17026 255 255 255 255 128 57825 225 24415 95 194 0 8417 0 16289 0 41377 0 49664 0 49408 150 49408 194 25088 225 0 255 65 255 129 255 161 159 192 127 53970 210 20303 79 129 0 160 0 17026 0 33153 0 32768 0 40704 130 33280 128 16384 128 0 128 34 161 65 194 128 128 130 64 49344 192 16448 64 45520 161 41440 117 45266 106 49856 124 49538 104 49281 151 49791 188 45681 207 40863 224 41408 224 40930 222 37359 235 40930 200 45746 178 12079 47 24703 79 25249 82 25216 16 33410 63 25151 31 24892 62 24631 94 16656 96 24674 161 16738 129 12640 129 8544 98 8546 82 41634 162 0 0 0 824 984 428 154 153 155 151 7 0 8 7751 24 7751 32 7751 40 7751 16 7751 48 7751 0 0 .sdw 13 145 0 [frm] 5 537395328 2801 4248 6207 4949 0 1 3 0 0 0 0 0 0 0 0 16777215 3 8 1361 3406 368 [frmname] Frame3 [frmlay] 4949 3406 1 0 0 1 4248 0 0 2 0 0 0 0 1 2801 6207 0 [isd] .X3 .tex .X3 1 1 0 0 3369 64882 100 0 0 .tex 0 65168 0 [lay] Standard 513 [rght] 15840 12240 1 1440 1440 1 1440 1440 0 1 0 1 0 2 1 1440 10800 12 1 720 1 1440 1 2160 1 2880 1 3600 1 4320 1 5040 1 5760 1 6480 1 7200 1 7920 1 8640 [hrght] [lyfrm] 1 11200 0 0 12240 1440 0 1 3 1 0 0 0 0 0 0 0 0 1 [frmlay] 1440 12240 1 1440 72 1 792 1440 0 1 0 1 1 0 1 1440 10800 2 2 4680 3 9360 [txt] <+B><:f160,,>VII. The Origion of Disorder in <:f160,2Symbol,0,0,0>g<:f160,,>-Al<+'>2<-'>O<+'>3<-'><:f> > [frght] [lyfrm] 1 13248 0 14400 12240 15840 0 1 3 1 0 0 0 0 0 0 0 0 2 [frmlay] 15840 12240 1 1440 792 1 14472 1440 0 1 0 1 1 0 1 1440 10800 2 2 4680 3 9360 [txt] <+B><:s><:f160,,> <+B><:s><:f160,,> <+B><:f160,,><:P10,0,VII - > > [elay] [l1] 0 [pg] 21 24 481 121 64 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 34 1939 49 32 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 51 2493 304 32 0 0 0 65535 2770 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 59 0 29 96 0 0 0 65535 2 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 73 2155 103 96 0 0 0 65535 2063 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 79 1518 31 32 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 115 0 98 0 0 0 0 65535 2 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 146 426 92 0 0 0 0 65535 341 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 179 0 111 0 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 199 545 127 64 0 0 0 65535 644 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 207 871 281 0 0 0 0 65535 1124 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 269 0 52 0 0 0 0 65535 2 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 305 0 29 32 0 0 0 65535 2 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 320 652 137 0 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 332 0 55 32 0 0 0 65535 2 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 340 29 5 0 0 0 0 65535 2 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 355 0 167 32 0 0 0 65535 80 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 365 0 29 32 0 0 0 65535 2 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 385 0 29 0 0 0 0 65535 2 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 419 0 34 32 0 0 0 65535 2 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 456 0 29 1025 0 0 0 65535 2 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 [edoc] <+B><:s><:#568,9360><:f480,,><+!> <+B><:s><:#568,9360><+!><:f480,,>VII<:f> <+B><:s><:#288,9360><+!> <+B><:s><:#288,9360><+!> <+B><:#592,9360><+!><:f320,,> <:f480,,>The Origin<-!><:f><+!><:f480,,> of Disorder in <:f480,2Symbol,0,0,0>g<:f480,,>-Al<+'>2<-'>O<+'>3<-!><-'><:f> <+@><:s><:#280,9360> <+@><:s><:#280,9360> <+@><:s><:#280,9360> <+@><:s><:#280,9360> <+@><:s><:#280,9360> <+@><:s><:#280,9360> <+@><:s><:#280,9360> <+@><:s><:#280,9360> <+@><:s><:#280,9360> <+@><:s><:#280,9360> <+@><:s><:#280,9360> <+@><:#376,9360><:f320,,><+!>VII.1 Introduction<-!><:f> <+@><:s><:#280,9360> <+@><:s><:#280,9360> <:#576,9360>The purpose of present work is to use computer simulation to understand better the structure of <:f240,2Symbol,0,0,0>g<:f>-Al<+'>2<-'>O<+'>3<-'> , in particular its <+">disordered<-"> nature. <:s><:#280,9360> It is widely accepted that <:f240,2Symbol,0,0,0>g<:f>-Al<+'>2<-'>O<+'>3<-'> has a spinel-like structure because the powder X-Ray peaks resemble those of real spinel. There are three features of importance which are clear from the data. A standard spinel pri mitive unit cell is cubic having 32 oxygen atoms forming fcc structure, with half of all octahedral (Oh) and 1/4 of tetrahedral (T) interstitial sites regularly occupied by cations. Thus the first and second features are that the oxygens have an fcc packing and that the Al atoms are distributed on both Oh and T sites. The structure of <:f240,2Symbol,0,0,0>g<:f>-Al<+'>2<-'>O<+'>3<-'> contrasts with that of <:f240,2Symbol,0,0,0>a<:f>-Al<+'>2<-'>O<+'>3<-'> in which the oxygens from an hcp structure and the Al are only in Oh sites and well ordered. The evidence for the occupation of both Oh and T sites by Al ions in <:f240,2Symbol,0,0,0>g<:f>-Al<+'>2<-'>O<+'>3<-'> comes from IR and Raman vibrational spectra which are sensitive to the local atomic structure. The third point is that there is a substantial degree of disorder, within the spinel-like short and medium range order. This is indicated by the broadening of the X-ray powder diffraction peaks. It also follows form the fact that the spinel unit cell with 32 oxygen ions requires 24 cations, but Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f>O<+'>3<-'> correspond only to Al<+'>21.33<-'>O<+'>32<-'>. Indeed this may be the cause of the disorder because the strong Al<+&>3+<:f240,2Symbol,0,0,0><-&>-<:f>Al<+&>3+<-&> repulsion would give a substantially different Al distribution, not just a few vacancies in the spinel structure. <:s><:#280,9360> The main questions are : why are both T and Oh sites occupied in <:f240,2Symbol,0,0,0>g<:f>-Al<+'>2<-'>O<+'>3<-'> compared with only Oh sites in <:f240,2Symbol,0,0,0>a<:f>-Al<+'>2<-'>O<+'>3<-'> when the underlying oxygen structures are so similar, namely fcc and hcp ? Is the occupation of T sites by Al stable in fcc framework ? Or do they just reflect the structure that <:f240,2Symbol,0,0,0>g<:f>-Al<+'>2<-'>O<+'>3<-'> inherited from its parent material and would they finally transform into Oh sites ? Even given the fact that both T and Oh sites are occupied, why are they disordered in <:f240,2Symbol,0,0,0>g<:f>-Al<+'>2<-'>O<+'>3<-'> ? <+">Ab initio<-"> calculations, by which we mean solving the Schr<\v>dinger equation for the whole system, are used in the present study. The reason is that there has been a bad experience in our research group : we have seen slightly different empirical sh ell models, both fitted to structures and dielectric data, giving substantially different results for the energy difference between Al on T and Oh sites. Therefore an <+">ab initio<-"> approach is necessary because in <:f240,2Symbol,0,0,0>g<:f>-Al<+'>2<-'>O<+'>3<-'> the occupation of both Oh and T sites by Al ions appears to be a crucial aspect of its structure. However, <:f240,2Times New Roman,0,0,0>the disordered/non-stoichiometric nature of<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<+'>3<-'><:f><:f240,2Times New Roman,0,0,0> makes<:f><:f240,2Times New Roman,0,0,0> a full super-cell <+">ab initio<-"> calculation <:f><:f240,2Times New Roman,0,0,0>impractical. For example, it would require 3 times the spinel unitcell to have an integer<:f><:f240,2Times New Roman,0,0,0> number of Al ions in a supercell. <:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0>With such a large supercell of 160 atoms, or similar sized cell, it would be too expensive to perform many calculations with the Al ions in different positions to find the ground state.<:f> <:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#864,9360>Since an investigation of <:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<+'>3<-'><:f><:f240,2Times New Roman,0,0,0> by complete <+">ab initio<-"> <:f>set of calculations is impossible, we use a 3-step approach instead. The idea is that we can represent the total energy of the system <+">E<-"> in terms of a model Hamiltonian to give the energy expression in the form <:s><:#280,9360> <:#296,9360>(1.1) <:f240,2Times New Roman,0,0,0><+">E <-"><:f>= <+">E<-"><+'>0<-'> + (<:f240,2Symbol,0,0,0>m<:f>/2) (N<+'>Oh<-'> <:f240,2Symbol,0,0,0>-<:f> N<+'>T<-'>) + <:f240,2Symbol,0,0,0>S<:f> <+">J<-"><:f240,2Times New Roman,0,0,0><+'>ij<-'><:f> <:f240,2Symbol,0,0,0>h<+'><:f240,2Times New Roman,0,0,0>i<-'> <:f><:f240,2Symbol,0,0,0>h<+'><:f240,2Times New Roman,0,0,0>j<-'><:f><-'> <:s><:#280,9360> in which <+">E<-"><+'>0<-'> is the energy constant, <:f240,2Symbol,0,0,0>m<:f> is the chemical potential energy difference of one Al ion on an Oh site rather than a T site, N<+'>Oh<-'> and N<+'>T<-'> are the numbers of Al on Oh and T sites, and <:f240,2Symbol,0,0,0>h<:f><+'>i<-'> is equal to unity when site {i} is occupied and zero when empty. The <+">J<-"><:f240,2Times New Roman,0,0,0><+'>ij<-'><:f> is the pair-wise interaction between Al atoms on site i and j, including both Oh and T type sites. We have used a simplified form of this model Hamiltonian in our study, which will be discussed later in Section 4. The reasons to introduce it here in the form of (1.1) is that it high-lights assumption that the interaction is pair-wise as shown by the term <:f240,2Symbol,0,0,0>S<:f> <+">J<-"><:f240,2Times New Roman,0,0,0><+'>ij<-'><:f> <:f240,2Symbol,0,0,0>h<+'><:f240,2Times New Roman,0,0,0>i<-'> <:f><:f240,2Symbol,0,0,0>h<+'><:f240,2Times New Roman,0,0,0>j<-'><:f><-'>. With this from of model Hamiltonian, we carry out the following three steps : (1) We perform 10 <+">ab initio<-"> calculations on a sufficiently small cell to obtain the 10 energies for 10 different configurations of the Al atoms. (2) We fit the parameters <+">E<-"><+'>0<-'>, <:f240,2Symbol,0,0,0>m<:f>, and <+">J<-"><:f240,2Times New Roman,0,0,0><+'>ij<-'><:f> in the model Hamiltonian to the 10 <+">ab initio<-"> energies. (3) We then do larger scale simulations based on the fitted Hamiltonian to find the ground state distribution of Al atoms in <-'><-'><-'><-'><-'><-'><-'><:f240,2Times New Roman,0,0,0> <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<+'>3<-'><:f><:f240,2Times New Roman,0,0,0> (and <:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Symbol,0,0,0>a<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<+'>3<-'><:f><:f240,2Times New Roman,0,0,0> for comparison<:f><:f240,2Times New Roman,0,0,0>).<:f><:f240,2Times New Roman,0,0,0><:f> We found that the large scale simulations (step 3 above) reproduced the fact that the Al atoms only occupy Oh sites in <:f240,2Symbol,0,0,0>a<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<+'>3<-'><:f><:f240,2Times New Roman,0,0,0>. They also predicted a N<+'>Oh<-'> : N<+'>T<-'> <:f>ratio around 73% : 27% for the Oh and T cation distribution in <:f240,2Times New Roman,0,0,0><:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<+'>3<-'><:f><:f240,2Times New Roman,0,0,0>,<:f> which was later confirmed by solid state NMR measurements (giving 70% : 30%) done by colleagues in Chemistry Department at our suggestion. These two quantitative successes of model (1.1) support the belief that <:f240,2Times New Roman,0,0,0>the approximations made in the various steps of our<:f><:f240,2Times New Roman,0,0,0> computational study fulfil<:f><:f240,2Times New Roman,0,0,0> the accuracy needed to<:f><:f240,2Times New Roman,0,0,0> address<:f><:f240,2Times New Roman,0,0,0> the central question of<:f><:f240,2Times New Roman,0,0,0> this chapter<:f240,2Times New Roman,0,0,0>, namely the origin of disorder in <:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<+'>3<-'>. The simulations of<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<+'>3<-'><:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>in fact<:f><:f240,2Times New Roman,0,0,0> always resulted in similar disordered structures. By<:f> comparing the simulatation results for both <:f240,2Symbol,0,0,0>a<:f> (hcp) and <:f240,2Symbol,0,0,0>g<:f> (fcc) alumina based on the same model Hamiltonian, we show that the differences between them are due to the underlying geometry of the oxygen frameworks. The methodology we have established not only provides an insight into the fundamental difference between the <:f240,2Symbol,0,0,0>a<:f> and <:f240,2Symbol,0,0,0>g<:f> phases of Al<+'>2<-'>O<+'>3<-'> but also the structure and stability of the Al distribution in the "weakly ordered" material <:f240,2Symbol,0,0,0>g<:f>-Al<+'>2<-'>O<+'>3<-'>. <:s><:#280,9360> <:#1720,9360>In the following sections of this chapter, we will mention some background information on <:f240,2Symbol,0,0,0>g<:f>-Al<+'>2<-'>O<+'>3<-'> in Section 2. In Section 3 the details of the <+">ab initio<-"> calculations will be given and, in Section 4 the construction and fitting of the model Hamiltonian. Section 5 consists of the results from the simulations and corresponding NMR measurements. We will discuss what we have learnt in Section 6 and draw smme brief conclusions. Finally in Section 7 some possible plans for future studies will be very briefly mentioned which provide a wider scope for our interest in <:f240,2Symbol,0,0,0>g<:f>-Al<+'>2<-'>O<+'>3<-'>. <:s><:#280,9360> <:s><:#280,9360> <:s><:#280,9360> <:s><:#280,9360> <:s><:#280,9360> <:s><:#280,9360> <:#280,9360> <:#376,9360><:f320,,><+!>VII.2<-!><:f><+!><:f320,,> Background<-!><:f><+!><:f320,,> Experimental<-!><:f><+!><:f320,,> Information<-!><:f> <:s><:#280,9360> <:s><:#280,9360> <:#1136,9360>Before starting our analysis in the next section, we review briefly in this section the reasons why <:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<:f240,2Times New Roman,0,0,0><+'>3<-'><:f> is interesting, which, in fact, motivate our current and future research on this materia l. We will also mention some experimental facts that are important for our computational investigation or the understanding of this material.<-'><-'> <:s><:#280,9360> <:#3144,9360>The present study was initiated by Dr. W. Macrodt when he was at ICI. <:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<+'>3<-'> is important in the chemical industry as a catalyst support and, less frequently, as a catalyst itself. <:f>It <-!><:f240,2Times New Roman,0,0,0><-'><-'>is a good catalyst support because being a fine powder it has an enormously large effective surface area. Made from very fine particles (microcrystals)<:f><:f240,2Times New Roman,0,0,0>, it becomes a highly p orous<:f><:f240,2Times New Roman,0,0,0> solid when compressed and treated<:f><:f240,2Times New Roman,0,0,0>. With a very uniform distribution of microcrystal size, the porous solid has a fairly uniform width of channels, which makes the supported catalytic processes size-selective. <:f><:f240,2Times New Roman,0,0,0>The catalytic function of pure<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<+'>3<-'> <:f><:f240,2Times New Roman,0,0,0>itself comes from the formation of catalytic<:f><:f240,2Times New Roman,0,0,0> centres on its surface in the presence of a solvent<:f><:f240,2Times New Roman,0,0,0>. Even when <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<+'>3<-'><:f><:f240,2Times New Roman,0,0,0> is used as a support,<:f><:f240,2Times New Roman,0,0,0> people believe that the substrates/reactants will first diffuse on its surface<:f240,2Times New Roman,0,0,0> <-'><-'> before the catalytic reaction takes place : therefore in that way <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<+'>3<-'> <:f><:f240,2Times New Roman,0,0,0>also chemically takes parts in the catalytic reaction. We understand that it is also used in paints and in explosives.<-!><-!><-!><-'> <:#280,9360> <:f>We now turn to the background information relevant to the present study. Firstly, <:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<:f240,2Times New Roman,0,0,0><+'>3<-'><:f><:f240,2Times New Roman,0,0,0> appears<:f><:f240,2Times New Roman,0,0,0> easily<:f><:f240,2Times New Roman,0,0,0> in many kinds<:f><:f240,2Times New Roman,0,0,0> of process involved in the formation of alumina such as the dehydration of AlOOH <[><:f240,2Times New Roman,0,0,0>Ref.G.1<:f240,2Times New Roman,0,0,0>] and the MBE growth of alumina films <[><:f240,2Times New Roman,0,0,0>Ref.C.2<:f240,2Times New Roman,0,0,0> ] as an intermediate step,<:f><:f240,2Times New Roman,0,0,0> finally leading to the most stable <:f240,2Symbol,0,0,0>a<:f240,2Times New Roman,0,0,0> (corundum) form after annealing<:f><:f240,2Times New Roman,0,0,0>.<:f><:f240,2Times New Roman,0,0,0> Thus one might expect<:f><:f240,2Times New Roman,0,0,0> <:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<:f240,2Times New Roman,0,0,0><+'>3<-'><:f240,2Times New Roman,0,0,0> to have a lower nucleation energy cost than that of <:f240,2Symbol,0,0,0>a<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<:f240,2Times New Roman,0,0,0><+'>3<-'><:f240,2Times New Roman,0,0,0>, and it would be interesting<:f><:f240,2Times New Roman,0,0,0> to know why. (Is the surface energy lower than that of <:f><:f240,2Symbol,0,0,0>a<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<:f240,2Times New Roman,0,0,0><+'>3<-'><:f><:f240,2Times New Roman,0,0,0> , or is it a volume effect ?) <:f><:f240,2Times New Roman,0,0,0>It was only recently that single-crystal <:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<:f240,2Times New Roman,0,0,0><+'>3<-'><:f240,2Times New Roman,0,0,0> has been claimed to have been successfully prepared by <:f><:f240,2Times New Roman,0,0,0>MOMBE growth on <:f><:f240,2Times New Roman,0,0,0>a specially <:f><:f240,2Times New Roman,0,0,0>selected<:f><:f240,2Times New Roman,0,0,0> substrate<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0><[><:f240,2Times New Roman,0,0,0>Ref.I.1<:f240,2Times New Roman,0,0,0>]. Such a highly sophisticated way for single-crystal<:f><:f240,2Times New Roman,0,0,0> preparation, together<:f><:f240,2Times New Roman,0,0,0> with the long term puzzle<:f><:f240,2Times New Roman,0,0,0> of the difficulty in growing bigger single crystals using industrial methods, high-lights the interesting nature of the crystallinity of <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<:f240,2Times New Roman,0,0,0><+'>3<-'><:f><:f240,2Times New Roman,0,0,0>.<:f><:f240,2Times New Roman,0,0,0> <:#2848,9360><:f240,2Times New Roman,0,0,0>Secondly, <:f>the material exists almost exclusively as micro-crystals, as seen under <:f240,2Times New Roman,0,0,0>the <:f><:f240,2Times New Roman,0,0,0>Transmintion<:f><:f240,2Times New Roman,0,0,0> Electron Microscope<:f><:f240,2Times New Roman,0,0,0> (TEM) <[><:f240,2Times New Roman,0,0,0>Ref.R.1<:f240,2Times New Roman,0,0,0>].<:f> <:f240,2Times New Roman,0,0,0>The fine particles of <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<:f240,2Times New Roman,0,0,0><+'>3<-'><:f><-'> <:f240,2Times New Roman,0,0,0>are shown to be nearly hexagonal-shaped platelets. The size of <:f><:f240,2Times New Roman,0,0,0>individual<:f><:f240,2Times New Roman,0,0,0> small grains is typically<:f><:f240,2Times New Roman,0,0,0> a<:f><:f240,2Times New Roman,0,0,0> diameter<:f><:f240,2Times New Roman,0,0,0> of 10-20 nm<:f><:f240,2Times New Roman,0,0,0> and thickness<:f><:f240,2Times New Roman,0,0,0> less than 5 nm. <:f><:f240,2Times New Roman,0,0,0>Under High Resolution TEM<:f><:f240,2Times New Roman,0,0,0> (HRTEM) <[><:f240,2Times New Roman,0,0,0>Ref.R.1<:f240,2Times New Roman,0,0,0>],<:f><:f240,2Times New Roman,0,0,0> which<:f> <:f240,2Times New Roman,0,0,0>probes the local structure at atomic resolution<:f>,<:f240,2Times New Roman,0,0,0> regular atomic textures and clear facets <:f><:f240,2Times New Roman,0,0,0>not only provide the evidence of the crystallinity<:f><:f240,2Times New Roman,0,0,0> of <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<:f240,2Times New Roman,0,0,0><+'>3<-'><:f><-'> <:f240,2Times New Roman,0,0,0>but also allow one to measure the ratio of atomic layer spacing along different lateral directions. Tn this way the Miller Indices of<:f><:f240,2Times New Roman,0,0,0> the surfaces of the crystallites normal to electron beam ca n be determined. This analysis suggests that the preferentially exposed surface of <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<:f240,2Times New Roman,0,0,0><+'>3<-'><:f><-'> <:f240,2Times New Roman,0,0,0>is <[>110], which will be an interesting point to study computationally in the future.<:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0> <:#280,9360><:f240,2Times New Roman,0,0,0> Thirdly,<-'><-'><:f240,2Times New Roman,0,0,0> it is often quoted that <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<:f240,2Times New Roman,0,0,0><+'>3<-'><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>has a "defective <:f><:f240,2Times New Roman,0,0,0>cubic spinel structure" with lattice parameter a<+'>0<-'> = 7.92 <\E>. <[>Ref.C.2],<:f><:f240,2Times New Roman,0,0,0> as suggested by<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>powder X-ray <:f><:f240,2Times New Roman,0,0,0>diffraction<:f>.<:f240,2Times New Roman,0,0,0> The fact that it has an fcc oxygen framework is widely accepted<:f><:f240,2Times New Roman,0,0,0>. <:f><:f240,2Times New Roman,0,0,0>We will use these facts (fcc and a<+'>0<-'>) to construct the appropriate supercell for our <+">ab initio<-"> calculations. <:f><:f240,2Times New Roman,0,0,0>Although basically crystalline, <:f><:f240,2Times New Roman,0,0,0>there is also an element of disorder in <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<:f240,2Times New Roman,0,0,0><+'>3<-'><:f><:f240,2Times New Roman,0,0,0> as seen <:f><:f240,2Times New Roman,0,0,0>from the width<:f><:f240,2Times New Roman,0,0,0> of Bragg peaks in <:f><:f240,2Times New Roman,0,0,0>powder X-ray pattern<:f><:f240,2Times New Roman,0,0,0>. The oxygen<:f><:f240,2Times New Roman,0,0,0> lattice looks nearly perfect whereas the peaks from the Al atoms appear broader. Our own estimation (unpublished) suggests that the medium range order of the oxygen structure has a with correlation length extending 10 fcc lattice<:f><:f240,2Times New Roman,0,0,0> cells or more, whereas the Al peaks are 3 times broader so they have a short range order extending to about 3 fcc c ells. <:f><:f240,2Times New Roman,0,0,0>Some X-ray crystallographers<:f><:f240,2Times New Roman,0,0,0> describe <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<:f240,2Times New Roman,0,0,0><+'>3<-'><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>as "Weakly<:f><:f240,2Times New Roman,0,0,0> Ordered"<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0><[><:f240,2Times New Roman,0,0,0>Ref.S.2<:f240,2Times New Roman,0,0,0>]<:f><:f240,2Times New Roman,0,0,0> due to the existence<:f><:f240,2Times New Roman,0,0,0> of internal disorder within the overall <:f>crystalline order in this material. The link between cation disorder in <:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<+'>3<-'><:f> and its micro-crystalline nature will be another interesting aspect for us to investigate in the future.<-'> <:s><:#280,9360> Fourthly, <:f240,2Times New Roman,0,0,0>the existence of both Oh and T coordinated Al in <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<:f240,2Times New Roman,0,0,0><+'>3<-'><:f><:f240,2Times New Roman,0,0,0> is shown by <:f><:f240,2Times New Roman,0,0,0>IR and Raman<:f> <:f240,2Times New Roman,0,0,0>spectroscopies<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0><[>Ref.C.1]<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>which are sensitive to the symmetry of the local environment of the Al atoms <:f><:f240,2Times New Roman,0,0,0>because they detect the allowed transitions between different <:f><:f240,2Times New Roman,0,0,0>vibrational<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>modes according to their respective<:f><:f240,2Times New Roman,0,0,0> section rules. <:f><:f240,2Times New Roman,0,0,0>Analysing both the numbers of IR<:f><:f240,2Times New Roman,0,0,0> and Raman active modes <:f><:f240,2Times New Roman,0,0,0>of Al-O vibration, experimentalists<:f><:f240,2Times New Roman,0,0,0> have shown that their results are <:f><:f240,2Times New Roman,0,0,0>consistent with the assumption that modes with<:f><:f240,2Times New Roman,0,0,0> Oh-coordinated AlO<+'>6<-'> and T-coordinated AlO<+'>4<-'><:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>local symmetry <:f><:f240,2Times New Roman,0,0,0>both contribute to what has been measured. The IR and Raman data therefore support the co-existence of T-site and Oh-site occupation by Al in <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<:f240,2Times New Roman,0,0,0><+'>3<-'><:f><:f240,2Times New Roman,0,0,0>.<:f> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:f240,2Times New Roman,0,0,0>Fifthly,<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>directly monitoring<:f><:f240,2Times New Roman,0,0,0> the phase transition process<:f><:f240,2Times New Roman,0,0,0> by <:f><:f240,2Times New Roman,0,0,0>i<:f><:f240,2Times New Roman,0,0,0>n-situ TEM <:f><:f240,2Times New Roman,0,0,0><[>Ref.S.3]<:f><:f240,2Times New Roman,0,0,0> enables one to measure<:f><:f240,2Times New Roman,0,0,0> the growth<:f><:f240,2Times New Roman,0,0,0> rate of <:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0> the phase from the amorphous material from<:f><:f240,2Times New Roman,0,0,0> video-records of the speed of the b<:f><:f240,2Times New Roman,0,0,0>oundary between two phases, giving estimated activation en ergies of 1.6 eV (for T << 750C) and 7.8 eV (for T <;> 750C) <:f><:f240,2Times New Roman,0,0,0>if the simple assumption of a thermally activated<:f><:f240,2Times New Roman,0,0,0> process is valid. In another work, activation energies for bulk diffusion of O and Al in <:f240,2Symbol,0,0,0>a<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<:f240,2Times New Roman,0,0,0><+'>3<-'><:f240,2Times New Roman,0,0,0> were reported to be 6.6 eV and 5.5 <:f><:f240,2Times New Roman,0,0,0>eV respectively, and an activation energy of transformation from <:f240,2Symbol,0,0,0>a<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<+'>3<-'> to <:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<+'>3<-'> of about 3.6 eV. Although not rigorously<:f><:f240,2Times New Roman,0,0,0> consistent with each others, these data do provide the order of magnitude<:f><:f240,2Times New Roman,0,0,0> <:f>of the energies involved in these processes, which we will later find comparable with our computational results. <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:f240,2Times New Roman,0,0,0>In conclusion, the picture <:f><:f240,2Times New Roman,0,0,0>of <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<:f240,2Times New Roman,0,0,0><+'>3<-'><:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>we have is that it consists of micro-crystals<:f><:f240,2Times New Roman,0,0,0> with quite good crystallinity of the fcc oxygen lattice. The Al atoms are distributed in a similar way to the cations in spinel structure, occ upying both Oh and T sites, but with considerable disorder.<:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0> Further evidence of the disorder/inhomogeneity of the cation distribution in alumina comes from the asymmetric tunnelling experiments<:f><:f240,2Times New Roman,0,0,0>, which suggest that inhomogeneous<:f><:f240,2Times New Roman,0,0,0> distribution<:f><:f240,2Times New Roman,0,0,0> of Al in aluminium-oxide is responsible for an asymmetrical<:f><:f240,2Times New Roman,0,0,0> current-voltage characteristics<:f><:f240,2Times New Roman,0,0,0>. <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360> <:s><:#280,9360> <:s><:#280,9360> <-!><+!><:f320,2Times New Roman,0,0,0>VII.3 Configuration Energies from First Principles Calculations<:f> <:s><:#280,9360><-!><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:f240,2Times New Roman,0,0,0>The interaction between atoms in a complex solid contains many different effects, and the local chemical environment<:f><:f240,2Times New Roman,0,0,0> can also be very different even for the same kind of element. Thus empirical interatomic<:f><:f240,2Times New Roman,0,0,0> potentials which have been fitted from limited experimental<:f><:f240,2Times New Roman,0,0,0> data, have limited capability for describing a system in such a<:f><:f240,2Times New Roman,0,0,0> situation.<:f> To have <:f240,2Times New Roman,0,0,0>a more appropriate starting point <:f><:f240,2Times New Roman,0,0,0>we take the <+">ab initio<-"> approach<:f>, which is used first to understand the characteristics of our system. In practice we make the ab initio calculations to evaluate the total energies of a chosen set of configuration s, and then use these in Section 4 to determine the parameters in the model Hamiltonian. <:s><:#280,9360> Before embarking on the ab initio calculations, we give first a more detailed critique of the empirical interatomic potentials and our reason for rejecting them, The most sophisticated of the empirical potentials is the shell model in which an atom or ion i s modelled by an electrostatically changed core, surrounded by a moveable shell (also charged) connected by a spring to the core and interacting with the shell of neighbouring atoms <:f240,2Times New Roman,0,0,0><[>Ref.C.3,D.1]<:f>, The parameters are determined by fitting to a range of crystal structures including their lattice constants, some elastic constants and some dielectric constants. In Professor Volker Heine's opinion, attemp ts to model oxides as purely ionic materials have often led to inconsistencies <:f240,2Times New Roman,0,0,0><[>Ref H.1]<:f>. Their bonding involves a considerable degree of covalency, as evidenced for example by the bond bending terms at Al (and Si) atoms needed in modelling tetrahedral framework <:f240,2Times New Roman,0,0,0>alumino-silicates<:f> structures (Ref. used by Thaya)<:f240,2Times New Roman,255,0,255><:f>. In particaular the covalency effect will differ between Al in Oh and T environments and we therefore expect empirical shell models to be unreliable in representing the energy difference between them. This is crucial for the present study because the Oh and T energy difference is one of the key features in understanding <:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<:f240,2Times New Roman,0,0,0><+'>3<-'>. Indeed in another research<:f><:f240,2Times New Roman,0,0,0> project <:f240,2Times New Roman,0,0,0>(done by S. Padlewski)<:f240,2Times New Roman,0,0,0> involving energy difference between Oh and T Al in mullite, it was found that the results were very sensitive to the empirical potential model used and one reputable paramete risation gave complete nonsense (unpublished !). The point is that these shell model potentials have been fitted to structures and elastic properties, with no information in the database about the energy difference<:f><:f240,2Times New Roman,0,0,0> between Al in Oh and T sites. Such failure is not uncommon when using empirical interatomic potentials well outside the type of situation included in the data base to which they have been fitted. However, our research<:f><:f240,2Times New Roman,0,0,0> group has bee n associated with <+">ab initio<-"><:f><:f240,2Times New Roman,0,0,0> total energy calculation for solids since near their inception about 15 years ago and we know of no case where a calculation by a reputable group world-wide has had to be filed in the was te paper bin<:f><-'> (V. Heine, private communication), in contrast to the simulation with emperical potentials. <:s><:#280,9360> In an <+">ab initio<-"> calculation, a unit cell is chosen as large as needed or computationally possible to model a complicated process or structure, in our case different configurations of Al in Oh and T sites, which is repeated periodically over the enti re space. In such a way, the so-called supercell geometry is used to represent an infinite solid. The Schr<\v>dinger equation is the solved for all the valance electrons in the whole system which provide the interatomic bonding. We use the total energy pa ckage CASTEP to perform the <+">ab initio<-"> calculations <[>Ref.P.1]. The formalism is based on Kohn-Sham Scheme of Density Functional Theory (DFT), a special represtation of quantum mechanics, with the Local Density Approximation (LDA) for complicated ex change-correlation energy, which come from the many-body effect of interacting electrons. We expand wavefunctions in terms of planewaves which is convenient for handling the periodic boundary conditions. In stead of true potentials for the ions, pseudopoten tials are used so that only the valence electrons are treated explicitly in the numerical procedure. At the same time the highly oscillatory part of the wavefunctions within the ion core regions is smoothed away, which reduces enormously the number of plane waves needed to express these wavefunctions. Using pseudopotentials therefore makes the expensive computing affordable. In this regard, an important aspect of CASTEP is that it directly minimises the total energy without diagonalising the one-electron Hamil tonian. For a given ionic configuration, the electronic ground state can be found, its energy calculated, and forces on each ion evaluated. The ions can then be moved, or relaxed, to the positions such that the system reaches its (stable or metastable) tota l energy minimum of both ionic and electronic structures. Thus our <+">ab initio<-"> calculations are static ones, effectively at zero degree Kelvin. <:s><:#280,9360> The preparation of <+">ab initio<-"> calculations involves generating pseudopotentials, choosing the unit cell geometry and constructing the initial ionic positions for each set of configurations of the Al atoms. Qc-tuned optimised pseudopotentials of Al an d O are used, including for Al the use of the techniques developed in Chapter IV for approximating the higher <+">d <-">component of the pseudopotential in terms of <:f240,2Times New Roman,0,0,0> <:f240,2Times New Roman,0,0,0>the <+">p<-"><:f240,2Times New Roman,0,0,0>-potential<:f>. Also the <:f240,2Times New Roman,0,0,0>O pseudopotential was optimised with a big core-radius<:f><:f240,2Times New Roman,0,0,0> (1.8 a.u) to take advantage of the negatively charged nature of oxygen ion in an oxide system for and thus reduce further the number of pl anewaves needed. <:s><:#280,9360><:f240,2Times New Roman,0,0,0> To generate the <+">ab initio<-"> database for our study, we used <:f240,2Times New Roman,0,0,0>an orthorhombic unit cell with a = 5.599 <\E>, b = 4.849 <\E>, c = 6.857 <\E>, <:f240,2Symbol,0,0,0>a <:f240,2Times New Roman,0,0,0>=<:f240,2Symbol,0,0,0> b <:f240,2Times New Roman,0,0,0>=<:f240,2Symbol,0,0,0> g <:f240,2Times New Roman,0,0,0>= 90<+&>o<-&>, containing 12 oxygen atoms in the fcc structrues and 8 aluminium<:f240,2Times New Roman,0,0,0> <:f>atoms. This cell dimensions have been set to match the <:f240,2Times New Roman,0,0,0>experimentally observed<:f><:f240,2Times New Roman,0,0,0> <:f>atomic spacing of<:f240,2Times New Roman,0,0,0> <:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<+'>3<-'><:f> with the lattice vectors <+!>a<-!> and <:f240,2Times New Roman,0,0,0><+!>b<-!><:f> span the basel plane and <:f240,2Times New Roman,0,0,0><+!>c<-!> pointing<:f> along the direction of the conventional ABCABC stacking sequence of an fcc strcture. The<:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>10 different sets of Al configurations are carefully chosen<:f> with the number Al atoms occupying Oh sites varying in the whole range of possibilities from 0 to 8. We have checked these input sets to make sure that these i nitial configurations contain various types of Al-Al pairs and distances to represent the <+">J<-"><+'>ij<-'> in Equation (1.1) well. <:f240,2Times New Roman,0,0,0>It is important to mention that the size of the unit cell covers the second nearest Oh-Oh and T-T sites, which should be sufficient to cover the major effects of pair-wise Al-Al inter action.<:f> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#5616,9360><:f240,2Times New Roman,0,0,0>Results of <+">ab initio<-"> calculations with both fully (all ions) relaxed and partly<:f><:f240,2Times New Roman,0,0,0> (Al only)<:f><:f240,2Times New Roman,0,0,0> relaxed<:f><:f240,2Times New Roman,0,0,0> were obtained. From these results we noticed that <:f><:f240,2Times New Roman,0,0,0>the relaxation of Al positions are large,<:f><:f240,2Times New Roman,0,0,0> leaving the Al ions well spread<:f><:f240,2Times New Roman,0,0,0> out<:f><:f240,2Times New Roman,0,0,0> among the in terstitial sites. <:f><:f240,2Times New Roman,0,0,0>Some of them had even hopped away from their <:f><:f240,2Times New Roman,0,0,0>initial input <:f><:f240,2Times New Roman,0,0,0>sites <:f><:f240,2Times New Roman,0,0,0>during the relaxation, <:f><:f240,2Times New Roman,0,0,0>which indicates the strong electrostatic nature of the Al-Al repulsion<:f><:f240,2Times New Roman,0,0,0>.<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>The relaxed oxygen framework still has a distorted fcc geometry<:f><:f240,2Times New Roman,0,0,0>.<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>We also note that it is computationally much more expensive<:f><:f240,2Times New Roman,0,0,0> to relax the oxygen position, which can be rationalised from the fact that the electrons in our system are mainly around the oxyg en ions, so that moving an oxygen ion creates a much larger disturbance<:f><:f240,2Times New Roman,0,0,0> to the electronic structure than moving Al. <:f><:f240,2Times New Roman,0,0,0>Thus the energies of the 10 configurations used to fit our model Hamiltonian (1.1) only have the Al ions relaxed, keeping the oxygens exactly in the fcc structure.<:f><:f240,2Times New Roman,0,0,0> These 10 different initia l configurations<:f><:f240,2Times New Roman,0,0,0> relaxed into<:f><:f240,2Times New Roman,0,0,0> only<:f><:f240,2Times New Roman,0,0,0> 9 <:f><:f240,2Times New Roman,0,0,0>distinguishable <:f><:f240,2Times New Roman,0,0,0>sets<:f><:f240,2Times New Roman,0,0,0> because two of them relaxed into the same final configuration.<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>Since we want to <:f><:f240,2Times New Roman,0,0,0>explore the energies<:f><:f240,2Times New Roman,0,0,0> of the system near its ground state <:f><:f240,2Times New Roman,0,0,0>instead of highly unstable configurations<:f><:f240,2Times New Roman,0,0,0>, <:f><:f240,2Times New Roman,0,0,0>we are happy<:f><:f240,2Times New Roman,0,0,0> to see that <:f><:f240,2Times New Roman,0,0,0>some <:f><:f240,2Times New Roman,0,0,0><+">ab initio<-"><:f><:f240,2Times New Roman,0,0,0> calculations converged into the<:f><:f240,2Times New Roman,0,0,0> same structure<:f><:f240,2Times New Roman,0,0,0>. <:f><:f240,2Times New Roman,0,0,0>To get a feeling for these results, we present two typical sets of Al-only relaxed configurations in Table.3.1.<:f><:f240,2Times New Roman,0,0,0> From an analysis of the Al-O distances of a given Al to its neighbouring<:f><:f240,2Times New Roman,0,0,0> oxygen ions, we immediately<:f><:f240,2Times New Roman,0,0,0> know whether an Al ion is in Oh or T site. <:f><:f240,2Times New Roman,0,0,0>It is also worth mentioning that <:f><:f240,2Times New Roman,0,0,0>the<:f><:f240,2Times New Roman,0,0,0> Al relaxation<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>in Oh sites often results 3 short <:f><:f240,2Times New Roman,0,0,0>and 3 long <:f><:f240,2Times New Roman,0,0,0>Al-O bonds<:f><:f240,2Times New Roman,0,0,0>. The situation for Al to form<:f><:f240,2Times New Roman,0,0,0> 1 long <:f><:f240,2Times New Roman,0,0,0>and <:f><:f240,2Times New Roman,0,0,0>3 short<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>Al-O bonds<:f><:f240,2Times New Roman,0,0,0> in a T site is less frequent but it does happen sometimes.<:f><:f240,2Times New Roman,0,0,0> The further analysis of the ionic positions and the total energies will be done<:f><:f240,2Times New Roman,0,0,0> in the next section in more detail.<:f> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#280,9360><:f240,2Times New Roman,0,0,0> <+B><:#280,9360><:f240,2Times New Roman,0,0,0><[>Table.3.1] <+B><:#240,9360><:f200,2Times New Roman,0,0,0>Distances of Al ions to its first 10 nearest oxygen neighnours (in Angs.). <:f><:f200,2Times New Roman,0,0,0>We used this information to <+B><:#240,9360><:f200,2Times New Roman,0,0,0>identify whether an Al ion is in Oh or T <:f><:f200,2Times New Roman,0,0,0>sites, <:f><:f200,2Times New Roman,0,0,0>also note that the strong relaxation of Al positions in <+B><:#240,9360><:f200,2Times New Roman,0,0,0>Oh and <:f><:f200,2Times New Roman,0,0,0>T<:f><:f200,2Times New Roman,0,0,0> sites <:f><:f200,2Times New Roman,0,0,0>with <:f><:f200,2Times New Roman,0,0,0>tendency to form 3 short bonds. <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#288,9360>Case 1 (<:f240,2Times New Roman,0,0,0><+">ab initio<-"><:f> No.3) <:#280,9360> <:#160,9360><:f160,QCourier,> Oxygen neighbour Al(1) Al(2) Al(3) Al(4) Al(5) Al(6) Al(7) Al(8)<:f> <:#280,9360> <:#160,9360><:f200,QCourier,> 1-th O neighbor 1.65 1.74 1.72 1.76 1.70 1.78 1.81 1.75 <:#160,9360><:f200,QCourier,> 2-th O neighbor 1.71 1.79 1.74 1.77 1.70 1.79 1.81 1.78 <:#160,9360><:f200,QCourier,> 3-th O neighbor 1.74 1.98 1.76 1.78 1.71 1.96 1.84 1.84 <:#160,9360><:f200,QCourier,> 4-th O neighbor 1.76 2.02 2.27 2.23 1.72 2.03 2.14 2.17 <:#160,9360><:f200,QCourier,> 5-th O neighbor 3.22 2.19 2.28 2.23 3.26 2.18 2.17 2.21 <:#160,9360><:f200,QCourier,> 6-th O neighbor 3.23 2.22 2.30 2.24 3.27 2.19 2.17 2.23 <:#160,9360><:f200,QCourier,> 7-th O neighbor 3.23 3.16 2.94 3.02 3.27 3.18 3.13 3.07 <:#160,9360><:f200,QCourier,> 8-th O neighbor 3.26 3.19 3.27 3.30 3.27 3.22 3.31 3.27 <:#160,9360><:f200,QCourier,> 9-th O neighbor 3.26 3.40 3.29 3.31 3.28 3.41 3.35 3.33 <:#160,9360><:f200,QCourier,> 0-th O neighbor 3.27 3.43 3.32 3.32 3.28 3.42 3.35 3.36 <:#160,9360><:f200,QCourier,> <:#160,9360><:f200,QCourier,> Occupation Type T Oh Oh Oh T Oh Oh Oh <:#160,9360><:f200,QCourier,> <:#240,9360><:f200,,> <:#288,9360>Case 2 (<+">ab initio<-"> No.9) <:#280,9360> <:#160,9360><:f160,QCourier,> Oxygen neighbour Al(1) Al(2) Al(3) Al(4) Al(5) Al(6) Al(7) Al(8)<:f> <:#280,9360> <:#160,9360><:f200,QCourier,> 1-th O neighbor 1.60 1.62 1.67 1.63 1.65 1.66 1.70 1.69 <:#160,9360><:f200,QCourier,> 2-th O neighbor 1.63 1.63 1.68 1.72 1.68 1.69 1.71 1.71 <:#160,9360><:f200,QCourier,> 3-th O neighbor 1.65 1.80 1.73 1.72 1.70 1.72 1.74 1.72 <:#160,9360><:f200,QCourier,> 4-th O neighbor 2.07 1.82 1.76 1.78 1.82 1.79 2.32 2.35 <:#160,9360><:f200,QCourier,> 5-th O neighbor 2.95 3.13 3.21 3.20 3.16 3.20 2.34 2.35 <:#160,9360><:f200,QCourier,> 6-th O neighbor 2.96 3.14 3.24 3.20 3.18 3.21 2.35 2.37 <:#160,9360><:f200,QCourier,> 7-th O neighbor 2.98 3.22 3.24 3.23 3.19 3.23 2.87 2.84 <:#160,9360><:f200,QCourier,> 8-th O neighbor 3.21 3.22 3.24 3.24 3.23 3.24 3.26 3.26 <:#160,9360><:f200,QCourier,> 9-th O neighbor 3.23 3.24 3.28 3.25 3.26 3.27 3.28 3.27 <:#160,9360><:f200,QCourier,> 10-th O neighbor 3.26 3.25 3.28 3.26 3.29 3.29 3.30 3.29<:f> <:#160,9360><:f200,QCourier,> <:#160,9360><:f200,QCourier,> Occupation Type <:f><:f200,QCourier,> T T T T T T Oh Oh <:#280,9360> <:s><:#280,9360> <:s><:#280,9360> <:s><:#280,9360> <:s><:#280,9360> <:s><:#280,9360> <:s><:#280,9360> <:#376,9360><:f320,,><+!>VII.4 The Model Hamiltonian<-!><:f> <:s><:#280,9360> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#288,9360><:f240,2Times New Roman,0,0,0><+!>VII.4.1 Constructing a Simplified<-!><:f><+!><:f240,2Times New Roman,0,0,0> Model Hamilton<-!><:f> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#1408,9360><:f240,2Times New Roman,0,0,0>An efficient <:f><:f240,2Times New Roman,0,0,0>Hamiltonian<:f><:f240,2Times New Roman,0,0,0> of the type of (1.1) is important for us so that<:f><:f240,2Times New Roman,0,0,0> its parameters can be determined by using the least possible numbers of configurations (and total energies) obtained from<:f><:f240,2Times New Roman,0,0,0> <+">ab initio<-"> calculations<:f><:f240,2Times New Roman,0,0,0>. <:f><:f240,2Times New Roman,0,0,0>The<:f><:f240,2Times New Roman,0,0,0> form of <:f><:f240,2Times New Roman,0,0,0>the model Hamiltonian<:f><:f240,2Times New Roman,0,0,0> we shall use is based on such practical consideration, <:f><:f240,2Times New Roman,0,0,0>and to achieve it <:f><:f240,2Times New Roman,0,0,0>we have made three essential further simplifications<:f><:f240,2Times New Roman,0,0,0> of the Hamiltonian <:f><:f240,2Times New Roman,0,0,0>as the follows :<:f> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#848,9360><:f240,2Times New Roman,0,0,0>(1) The same interaction <+">J<-"><+'>ij<-'> is shared<:f><:f240,2Times New Roman,0,0,0> by Al ions in both Oh and T sites<:f><:f240,2Times New Roman,0,0,0>, which greatly reduces<:f><:f240,2Times New Roman,0,0,0> the number of parameters in our model Hamiltonian because it is not necessary to distinguish<:f><:f240,2Times New Roman,0,0,0> between Oh-Oh, Oh-T and T-T interactions<:f><:f240,2Times New Roman,0,0,0>.<:f> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#568,9360><:f240,2Times New Roman,0,0,0>(2) <:f><:f240,2Times New Roman,0,0,0>The <+">J<-"><+'>ij<-'> is assumed to be just a function of the distances between<:f><:f240,2Times New Roman,0,0,0> Al ions, using the actual relaxed<:f><:f240,2Times New Roman,0,0,0> r<+'>ij<-'> <:f><:f240,2Times New Roman,0,0,0>distance between them<:f><:f240,2Times New Roman,0,0,0> instead of the site labelling<:f><:f240,2Times New Roman,0,0,0> (i,j). <:#848,9360><:f240,2Times New Roman,0,0,0>We therefore write <+">J<-"><+'>ij<-'> = <+">J<-">(<:f><:f240,2Times New Roman,0,0,0>r<+'>ij<-'><:f><:f240,2Times New Roman,0,0,0>)<:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0>. By doing this we further reduce the number of parameter needed in the Hamiltonian because we have avoided the complication of site labelling<:f><:f240,2Times New Roman,0,0,0> in a distorted/relaxed system. <:s><:#288,9360><+"><:f240,2Times New Roman,0,0,0> <:#1128,9360>(3) To avoid building in an unwarranted assumption about the form of the function <+">J<-">(r<+'>ij<-'>), we represent it by a series of steps covering the ranges 2.6 0.2 <\E>, 3.0 0.2 <\E>, ... etc. This helps to reduce the number of parameters further. The partitioning of the distance into different ranges can be adjusted to match the accuracy required. <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#848,9360><:f240,2Times New Roman,0,0,0>Our chosen simplifications of the model Hamiltonian<:f><:f240,2Times New Roman,0,0,0> may be justified<:f><:f240,2Times New Roman,0,0,0> to some extend by <:f><:f240,2Times New Roman,0,0,0>analysing the <+">ab initio<-"> results. In particular <:f><:f240,2Times New Roman,0,0,0>there seems to be sufficient space for the Al ions to relax away from the centres of both Oh and T cages. <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#2808,9360><:f240,2Times New Roman,0,0,0>This indicates that the radius of the Al ion is sufficiently small not to create large volume strains in either Oh or T sites : such volume strains, if present, would<:f><:f240,2Times New Roman,0,0,0> have given substantial<:f> <:f240,2Times New Roman,0,0,0> long range interactions. A significant volume strain<:f><:f240,2Times New Roman,0,0,0> would presumably be larger for the tetrahedral cage then for the octahedral one, thus making it necessary to distinguish between Oh and T s ites. The relaxation of the Al inside the tetrahedral or octahedral cage of oxygen ions also shows that a pure site labelling<:f><:f240,2Times New Roman,0,0,0> i,j for the interaction <+">J<-"><+'>ij<-'><+'><-'> would be extremely complicated because it would in principle<:f><:f240,2Times New Roman,0,0,0> have to take into account the displacement at each site. The fact that an Al ion hopped to a neighbouring site away from one another d uring the relaxation when two Al ion were close together suggested that their interaction is mainly electrostatic, which gives some further justification to not distinguishing between Oh and T site. <+'><-'> <:#280,9360><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0> <:#560,9360><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0>With these simplifications we can re-write our model<:f><:f240,2Times New Roman,0,0,0> Hamiltonian (1.1) explicitly <:f><:f240,2Times New Roman,0,0,0>, <:f>for a given cation configuration,<:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>as<:f><:f240,2Times New Roman,0,0,0>:<:f> <:s><:#280,9360> <:s>(4.1) <:A8> <:s><:#280,9360> <:#1728,9360>where r<+'>m<-'> is the midpoints 2.6 <\E>, 3.0 <\E>, ... representing the ranges 2.6 0.2 <\E>, 3.0 0.2 <\E> ... chosen to partition the Al-Al distances and n(r<+'>m<-'>) the number of pairs falling in that range with interaction <+">J<-"><-'>(r<+'>m<-'>). We have also transformed slightly the terms involving <:f240,2Symbol,0,0,0>m<:f>, using the fact that the total number of Al atoms N<+'>Oh<-'>+<+'> <-'><+'><-'>N<+'>T<-'><-'><-'> in a given simulation is constant, which also gives <-'><+"><:f240,2Times New Roman,0,0,0>E<:f240,2Times New Roman,0,0,0><-"><+'>0<-'>'<:f> in (4.,1) slightly different from <+">E<-"><+'>0<-'> in (1.1). The eight parameters <+">E<-"><+'>0<-'>' , <:f240,2Symbol,0,0,0>m<:f> and <:f240,2Times New Roman,0,0,0>(six) <+">J<-"><:f>(r<:f240,2Times New Roman,0,0,0><+'>m<-'><:f>) in (4.1) are determined from the <+">ab initio<-"> calculations by the procedure described in the next section. <:#576,9360>The reduction of the model Hamiltonian to eight parameters clearly minimises the number of <:f240,2Times New Roman,0,0,0><+">ab initio<-"><:f> calculations needed to fit them.<-'><-'><-'><-'> <:s><:#280,9360> <:#280,9360> <:#288,9360><:f240,2Times New Roman,0,0,0><+!>VII.4.2 Determine The Parameters in Model Hamiltonian<-!><:f> <:#280,9360> <:#1408,9360>The method we used was to fit the eight parameters uniquely to eight of <+">ab initio<-"> energies, omitting one of the nine energies in turn. In two cases this gave absured results which we neglected (Table 4.2), leaving the seven fits also listed in Table 4.2. There are very large differences between them and we also show the average of the seven fits, giving the parameters used in the subsequent simulations with their standard deviations. <:#280,9360> <+B><:#280,9360><:f240,2Times New Roman,0,0,0><[>Table.4.1] <+B><:s><:#280,9360><:f240,2Times New Roman,0,0,0> <+B><:#280,9360><:f240,2Times New Roman,0,0,0>Fitted parameters of model Hamiltonian (4.1)<:f> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#248,9360><:f200,QCourier,0,0,0> Parameters (eV) <:f><:f200,2Symbol,0,0,0>m<:f><:f200,QCourier,0,0,0> J(2.6) J(3.0) J(3.4) J(3.8) J(4.2) J(4.6) E<+'>0<-'>' <:s><:#160,9360><:f200,QCourier,0,0,0> <:s><:#160,9360><:f200,QCourier,0,0,0> Fit 1 -2.8 -2.6 <:f><:f200,QCourier,0,0,0> 0.2 -1.0 -0.7 -0.5 -0.7 -5566.2 <:s><:#160,9360><:f200,QCourier,0,0,0> Fit 2 -1.8 -0.1 -0.3 -0.6 -0.6 -0.4 -0.4 -5582.4 <:s><:#160,9360><:f200,QCourier,0,0,0> Fit 3 -0.1 1.6 0.4 0.0 0.1 -0.2 -0.1 -5623.3 <:s><:#160,9360><:f200,QCourier,0,0,0> Fit 4 -0.3 -1.3 -0.7 -0.5 0.0 -1.1 -0.3 -5583.6 <:s><:#160,9360><:f200,QCourier,0,0,0> Fit 5 -1.4 5.5 4.6 3.5 2.2 1.1 0.5 -5768.3 <:s><:#160,9360><:f200,QCourier,0,0,0> Fit 6 -1.9 1.9 2.1 1.0 0.8 -0.6 -0.3 -5652.8 <:s><:#160,9360><:f200,QCourier,0,0,0> Fit 7 -0.5 -1.1 -1.3 -1.7 -1.3 -0.8 -0.2 -5553.7 <:s><:#160,9360><:f200,QCourier,0,0,0> <:s><:#160,9360><:f200,QCourier,0,0,0> Average <:f200,QCourier,0,0,0>-1.26 0.56 0.71 0.10 0.07 -0.35 -0.21 -5619.04 <:#160,9360><:f200,QCourier,> <:s><:#248,9360><:f200,QCourier,189,191,193> <:f200,QCourier,0,0,0> <:f200,2Symbol,0,0,0>s<:f200,QCourier,0,0,0> 0.92 2.50 1.86 1.59 1.07 0.65 0.34 68.80<:f> <:#160,9360><:f200,QCourier,0,0,0> <:s><:#160,9360><:f200,QCourier,0,0,0> Discarded -15.7 43.0 37.8 21.8 13.9 1.2 -4.8 -6515.1 <:s><:#160,9360><:f200,QCourier,0,0,0> Discarded 7.8 -6.0 -3.1 -6.3 3.4 2.7 1.9 -5621.3<:f> <:s><:#280,9360> <:s><:#280,9360> <:#1136,9360><:f240,2Times New Roman,0,0,0>Again we make the point, <:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0>firstly, that the results seems to be sufficient for our limited purpose of understanding of the str ucture of <:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<+'>3<-'> as show below. Secondly the ab initio calculations are computationally very expensive and we are near the limit of what is feasible with moderate resources. <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#2272,9360><:f240,2Times New Roman,0,0,0>We note from the table that the chemical potential difference <:f240,2Symbol,0,0,0>m<:f> comes out negative i.e., the Oh site is favoured over the T one, as expected We also see from the table that <:f240,2Times New Roman,0,0,0>the Al-Al interaction (electrostatic energy) is comparable with the chemical potential difference of an Al in Oh site and in Td site, which is important because <:f><:f240,2Times New Roman,0,0,0>it means the Al-Al interaction may sometimes overcome the chemical potential <:f240,2Symbol,0,0,0>m<:f240,2Times New Roman,0,0,0> to favour having an Al ion in a T site rather than an Oh site. We are also happy to see that the Al-Al interaction is reasonable long<:f><:f240,2Times New Roman,0,0,0> (medium) ranged because otherwise th ere is no way for the Al ions to distinguish the fcc and hcp structures because<:f><:f240,2Times New Roman,0,0,0> of their similar local environments of the Oh and T sites.<:f> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#2256,9360><:f240,2Times New Roman,0,0,0>We also note that the stanard deviation in the average fit is<:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0> larger for<:f><:f240,2Times New Roman,0,0,0> small Al-Al distances,<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>but we are actually not too worried about that<:f><:f240,2Times New Roman,0,0,0> because it is<:f><:f240,2Times New Roman,0,0,0> fairly clear from the relaxed structures, e.g. from the information<:f><:f240,2Times New Roman,0,0,0> in Table 4.1, that the Al ions are highly<:f><:f240,2Times New Roman,0,0,0> repulsive to one another, so that at least the sign must be positive, larger error corresponding to this large repulsion is unlikely to change the behaviour<:f><:f240,2Times New Roman,0,0,0> of the Hamiltonian qualitatively. We will also see in the Sections 4 that the detail profile of the interaction play a less important role in the difference<:f><:f240,2Times New Roman,0,0,0> between <:f240,2Symbol,0,0,0>a<:f240,2Times New Roman,0,0,0> and <:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0> phases of Al<+'>2<-'>O<:f240,2Times New Roman,0,0,0><+'>3<-'><:f><:f>, and that our fit of interaction, although not very well, is already sufficie nt to reveal the main physical feature of the systems. <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#1688,9360><:f240,2Times New Roman,0,0,0>When the fitted<:f><:f240,2Times New Roman,0,0,0> model <:f><:f240,2Times New Roman,0,0,0>Hamiltonian (4.1) is used in a simulation, we have already cut-off the effect of interaction longer than 4.8 <\E>. because of the definition of (4.1)<:f><:f240,2Times New Roman,0,0,0>. <:f><:f240,2Times New Roman,0,0,0>As for the interaction ranges between <:f><:f240,2Times New Roman,0,0,0> 0 and 2.4 <\E>, we will assign an<:f><:f240,2Times New Roman,0,0,0> infinity (or computationally, extremely large) repulsion<:f><:f240,2Times New Roman,0,0,0> to represent the fact that <:f><:f240,2Times New Roman,0,0,0>having two Al ions such close is <:f240,2Times New Roman,0,0,0>energetically <:f240,2Times New Roman,0,0,0>highly unfavourable, <:f><:f240,2Times New Roman,0,0,0>since<:f><:f240,2Times New Roman,0,0,0> there is no one case in our <:f240,2Times New Roman,0,0,0><+">ab initio<-"><:f240,2Times New Roman,0,0,0> calculations that a relaxed Al-Al pair have a distance smaller than 2.4 <\E>, even though there initial conditions do allow that. <:f> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360> <:#280,9360> <:#376,9360><:f320,2Times New Roman,0,0,0><+!>VII.5 Simulation of Structures and Comparison with Experiments<-!><:f> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:f240,2Times New Roman,0,0,0>Computer simulations based on the fitted model Hamiltonian<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>(4.1) can now be used to investigate many properties of <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<+'>3<-'> <:f><:f240,2Times New Roman,0,0,0>such as <:f><:f240,2Times New Roman,0,0,0>the nature of the ground state, including short range and long range order, etc., on a much larger scale than can be achieved <+">ab initio<-">. <:f><:f240,2Times New Roman,0,0,0>The planned simulations are partly completed and we present here the results. We will attempt understand from them the<:f><:f240,2Times New Roman,0,0,0> disordered nature of the material in terms of occup ation of T-sites by Al and the occupation ratio to Oh-sites :<:f><:f240,2Times New Roman,0,0,0> also the effect of different oxygen frameworks (fcc, hcp) on such ratio. <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#3416,9360><:f240,2Times New Roman,0,0,0>The idea our of simulation is simple. <:f><:f240,2Times New Roman,0,0,0>We want to see what kind of Al distribution on Oh and T sites has the lowest total energy, which should tell us something about the structure of <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<+'>3<-'>. We also want to how the underlying fcc or hcp nature of the oxygen<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>framework affects the ground state distribution of cations.<:f> For these purposes, it is sufficient to model the Al<+'>2<-'>O<+'>3<-'> systems with interacting Al ions moving among all possible Oh and T interstitial sites of a given (fcc or hcp) oxygen framework. This naturally lead us to the choice of the Lattice Gas type simulation in which the Al ions are treated as gas particles interacting with each other according to <+">J<-">(r<+'>m<-'>), and hopping among the discretised <:f240,2Times New Roman,189,191,193>lattice points which represents the<:f> interstitial sites of the fcc or hcp structures, with an additional chemical potential <:f240,2Symbol,0,0,0>m<:f> on the Oh sites. We also assume that the hopping can only happen toward the nearest neighbouring sites. It is important to note that the potential energy of a given site can be evaluate by using <+">J<-">(r<+'>m<-'>) and <:f240,2Symbol,0,0,0>m<:f> from (4.1), so we know which are the lower energy sites to move Al ions to during the lattice gas simulation. <:s><:#280,9360> <:#3080,9360>There is a less trivial technical aspect that worth mentioning. It is not directly obvious how to construct the required "lattice sites" (not the lattice in crystallographers' sense) for our lattice gas simulation, in a simple and computationally addressabl e way, so that they preserve the correct connectivity of Oh and T interstitial sites. For example, in an fcc oxygen framework an Oh-site has 8 nearest neighbour sites, and a T-site has 4. If we represent all possible hops as lines between "lattice points", then we will see regular distribution of 8-arm vertices and 4-arm vertices. Their positions can not be described in terms of any common cubic or hexagonal structures. We have solved this problem by embedding the topology of the Oh and T sites into a denser simple cubic grid, which we found possible for both fcc and hcp cases. We could then express the connectivity between neighbouring interstitial sites in terms of lattice vectors, these vectors then defining all possible hopping directions from a site to its neighbours. <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#3080,9360><:f240,2Times New Roman,0,0,0>We have performed<:f><:f240,2Times New Roman,0,0,0> simulations<:f><:f240,2Times New Roman,0,0,0> on both fcc and hcp frameworks representing 864 oxygen ions. <:f><:f240,2Times New Roman,0,0,0>In each simulation, 576 Al "particles" are set to be hopping<:f><:f240,2Times New Roman,0,0,0> among corresponding lattice sites (864 Oh and 1728 T sites) in a box with<:f><:f240,2Times New Roman,0,0,0> periodic boundary co ndition<:f>.<:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>The simulation is performed in the way that Al "particles" will only hop to lowest energy neighbouring sites or no hop, therefore the total energy of the system will be reduced wit h such procedure. <:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0>Random noise is used occasionally<:f><:f240,2Times New Roman,0,0,0> to disturb<:f><:f240,2Times New Roman,0,0,0> the system to present it from stuck in some meta-stable<:f><:f240,2Times New Roman,0,0,0> state. The noise is not introduced through a temperature<:f><:f240,2Times New Roman,0,0,0> variable<:f><:f240,2Times New Roman,0,0,0>, though proper simulated annealing with a Boltzman factor can be carried out<:f><:f240,2Times New Roman,0,0,0> easily if one needs a more effective search of low energy states.<:f><:f240,2Times New Roman,0,0,0> The simulation continues until essentially all movement of Al stops, <:f>and then the final positions of these ions are analysed. <:f240,2Times New Roman,0,0,0>Various initial conditions, indicated in Table 4.2, were used to ensure a better exploration of structures<:f><:f240,2Times New Roman,0,0,0> with minimum<:f><:f240,2Times New Roman,0,0,0> total energy<:f><:f240,2Times New Roman,0,0,0>. <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:f240,2Times New Roman,0,0,0>The following table shows the results of searching in the above way for a low total energy Al population.<:f><:f240,2Times New Roman,0,0,0> In the first column "<:f200,QCourier,0,0,0>Initialisation<:f240,2Times New Roman,0,0,0> " we specify the initialisation condition for the simulation, in which "<:f200,QCourier,0,0,0>optimal<:f240,2Times New Roman,0,0,0>" means that a well separated initial Al configuration as ordered as possible<:f><:f240,2Times New Roman,0,0,0> has been used <:f><:f240,2Times New Roman,0,0,0>, "<:f><:f200,QCourier,0,0,0>condensed<:f><:f240,2Times New Roman,0,0,0>" means all Al ions were packed in a small region<:f><:f240,2Times New Roman,0,0,0> in the simulation box at the beginning<:f><:f240,2Times New Roman,0,0,0> , and "<:f200,QCourier,0,0,0>random<:f240,2Times New Roman,0,0,0>" means we randomly placed Al ions in all possible sites before starting. The words "<:f200,QCourier,0,0,0>Oh<:f240,2Times New Roman,0,0,0>", "<:f200,QCourier,0,0,0>T<:f240,2Times New Roman,0,0,0> " or "<:f200,QCourier,0,0,0>both<:f240,2Times New Roman,0,0,0>" appearing as prefixes in this column indicate<:f><:f240,2Times New Roman,0,0,0> whether<:f><:f240,2Times New Roman,0,0,0> the Al ions initially occupied Oh sites or T sites, or both. For exampl e, <:f><:f240,2Times New Roman,0,0,0>"<:f200,QCourier,0,0,0>Oh optimal<:f240,2Times New Roman,0,0,0>"<:f><:f240,2Times New Roman,0,0,0> means <:f><:f240,2Times New Roman,0,0,0>all Al ions started from well ordered Oh positions, well separated. <:f><:f240,2Times New Roman,0,0,0>The second column "<:f200,QCourier,0,0,0>n<[>Oh]/n<[>T]<:f240,2Times New Roman,0,0,0>" contains the number of Al ions in Oh and T sites respectively <:f><:f240,2Times New Roman,0,0,0>as the result of the simulation.<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>The last<:f><:f240,2Times New Roman,0,0,0> column<:f><:f240,2Times New Roman,0,0,0> "<:f200,QCourier,0,0,0>n<[>Oh]/n<[>Oh+T]<:f240,2Times New Roman,0,0,0>"<:f><:f240,2Times New Roman,0,0,0> gives the ratio of Oh sited<:f><:f240,2Times New Roman,0,0,0> Al as a fraction of all Al ions. The <:f>corresponding value from Solid State NMR measurements, which were carries out <+">after<-"> our simulation, as will be mentioned in more detail at the end of this section, are also listed. <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <+B><:s><:#280,9360><:f240,2Times New Roman,0,0,0><[>Table 5.1] <+B><:#280,9360><:f240,2Times New Roman,0,0,0>Results of simulations for Al in fcc and hcp oxygen structures with various initial conditions<:f><:f240,2Times New Roman,0,0,0>.<:f> <:s><:#240,9360><:f200,2Times New Roman,0,0,0> <:s><:#240,9360><:f200,2Times New Roman,0,0,0> <:f240,2Times New Roman,0,0,0>For the case of fcc oxygen framework :<:f> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#160,9360><:f200,QCourier,0,0,0>fcc <:s><:#160,9360><:f200,QCourier,0,0,0> ratio percentage <:s><:#160,9360><:f200,QCourier,0,0,0> <:s><:#160,9360><:f200,QCourier,>Initialisation n<[>Oh]/n<[>T] n<[>Oh]/n<[>Oh+T] <:s><:#160,9360><:f200,QCourier,> <:s><:#160,9360><:f200,QCourier,> Oh optimal 441/135 76% <:s><:#160,9360><:f200,QCourier,> Oh optimal 441/135 76% <:s><:#240,9360><:f200,,> <:s><:#160,9360><:f200,QCourier,> T optimal 408/168 70% <:s><:#160,9360><:f200,QCourier,> T optimal 415/161 72% <:s><:#160,9360><:f200,QCourier,> <:s><:#160,9360><:f200,QCourier,> T condense 428/148 75% <:s><:#160,9360><:f200,QCourier,> T condense 430/144 75% <:s><:#160,9360><:f200,QCourier,> <:s><:#160,9360><:f200,QCourier,> Oh random 414/162 72% <:s><:#160,9360><:f200,QCourier,> Oh random 411/165 71% <:s><:#160,9360><:f200,QCourier,> <:s><:#160,9360><:f200,QCourier,> T random 411/165 71% <:s><:#160,9360><:f200,QCourier,> T random 417/159 72% <:s><:#160,9360><:f200,QCourier,> <:s><:#160,9360><:f200,QCourier,> both random 411/165 71% <:s><:#160,9360><:f200,QCourier,> both random 432/144 75% <:s><:#160,9360><:f200,QCourier,> both random 419/157 73%<:f> <:s><:#160,9360><:f200,QCourier,> <:s><:#160,9360><:f200,QCourier,0,0,0>Predicted ratio <:f200,QCourier,0,0,0>73.0% <:s><:#160,9360><:f200,QCourier,0,0,0> <:s><:#160,9360><:f200,QCourier,0,0,0> <:s><:#160,9360><:f200,QCourier,0,0,0>Solid state NMR Result 70.66% +/- 2.28%<:f> <:s><:#160,9360><:f200,QCourier,0,0,0> <:s><:#160,9360><:f200,QCourier,0,0,0> <:s><:#160,9360><:f200,QCourier,0,0,0> <:s><:#160,9360><:f200,QCourier,0,0,0> <:s><:#160,9360><:f200,QCourier,0,0,0> <:f240,2Times New Roman,0,0,0>For the case of hcp oxygen framework : <:s><:#160,9360><:f200,QCourier,0,0,0> <:s><:#160,9360><:f200,QCourier,0,0,0>hcp <:s><:#160,9360><:f200,QCourier,0,0,0> ratio percentage<:f> <:s><:#160,9360><:f200,QCourier,0,0,0> <:s><:#160,9360><:f200,QCourier,>Initialisation n<[>Oh]/n<[>T] n<[>Oh]/n<[>Oh+T]<:f> <:s><:#160,9360><:f200,QCourier,0,0,0> <:s><:#160,9360><:f200,QCourier,> Oh optimal 576/0 100% <:s><:#160,9360><:f200,QCourier,> Oh optimal 576/0 100% <:s><:#160,9360><:f200,QCourier,> <:s><:#160,9360><:f200,QCourier,> Td optimal 576/0 100% <:s><:#160,9360><:f200,QCourier,> Td optimal 576/0 100% <:s><:#160,9360><:f200,QCourier,> <:s><:#160,9360><:f200,QCourier,> both condense 568/8 99% <:s><:#160,9360><:f200,QCourier,> both condense 568/8 99% <:s><:#160,9360><:f200,QCourier,> <:s><:#160,9360><:f200,QCourier,> Oh random 565/11 98% <:s><:#160,9360><:f200,QCourier,> Oh random 571/5 99% <:s><:#160,9360><:f200,QCourier,> Oh random 576/0 100% <:s><:#160,9360><:f200,QCourier,> Oh random 576/0 100% <:s><:#160,9360><:f200,QCourier,> <:s><:#160,9360><:f200,QCourier,> Td random 521/55 90% <:s><:#160,9360><:f200,QCourier,> Td random 576/0 100% <:s><:#160,9360><:f200,QCourier,> Td random 539/37 94% <:s><:#160,9360><:f200,QCourier,> Td random 539/37 94% <:s><:#160,9360><:f200,QCourier,> <:s><:#160,9360><:f200,QCourier,> both random 555/21 96% <:s><:#160,9360><:f200,QCourier,> both random 513/63 89% <:s><:#160,9360><:f200,QCourier,> both random 563/13 98% <:s><:#160,9360><:f200,QCourier,> both random 553/23 96% <:s><:#160,9360><:f200,QCourier,> both random 556/20 97%<:f> <:s><:#160,9360><:f200,QCourier,0,0,0> <:s><:#160,9360><:f200,QCourier,0,0,0>Predicted ratio 97.3%<:f> <:s><:#160,9360><:f200,QCourier,0,0,0> <:s><:#160,9360><:f200,QCourier,0,0,0> <:s><:#160,9360><:f200,QCourier,0,0,0>Solid state NMR result 100% +/- 0.00%<:f> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0>From the above results, we are pleased to find that : <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:f240,2Times New Roman,0,0,0>(1) The simulations are stable with respect to all those<:f><:f240,2Times New Roman,0,0,0> very different initial conditions. <:f><:f240,2Times New Roman,0,0,0>The results predicted an unambiguous n<[>Oh]/n<[>Oh+T] ratio of Al ions in <:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<+'>3<-'>,<:f><:f240,2Times New Roman,0,0,0> and in particular that the partial occupation of T sites is a energetically stable, not just a kinetic artefact<:f><:f240,2Times New Roman,0,0,0> t hat may come from the various parent materials during the preparation.<:f> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:f240,2Times New Roman,0,0,0>(2) The predicted percentage of n<[>Oh]/n<[>Oh+T] for <:f><:f240,2Symbol,0,0,0>g<:f>-Al<+'>2<-'>O<:f240,2Times New Roman,0,0,0><+'>3<-'><:f> agrees well with experiment, and the simulations also gives the correct ratio (all Oh Al) for <:f240,2Symbol,0,0,0>a<:f>-Al<+'>2<-'>O<:f240,2Times New Roman,0,0,0><+'>3<-'><:f>.<:f240,2Times New Roman,0,0,0> It is clear from our results that the difference between the fcc and hcp oxygen frameworks is responsible<:f><:f240,2Times New Roman,0,0,0> for whether the T-site are occupied by some Al ions. <:f> <:s><:#280,9360> <:f240,2Times New Roman,0,0,0>We want<:f><:f240,2Times New Roman,0,0,0> to emphasise that the simulations were carries out first so that the percentage of<:f><:f240,2Times New Roman,0,0,0> n<[>Oh] in <:f><:f240,2Symbol,0,0,0>g<:f>-Al<+'>2<-'>O<:f240,2Times New Roman,0,0,0><+'>3<-'><:f> was a prediction. Subsequently this ratio was measured at our suggestion with solid state NMR by colleagues Mr. Chi-Feng Cheng and Dr. Jacek Klinowski in the Chemistry De partment, Cambridge, with the results shown in Table 5.1. In spite of the problems of obtaining a good fit for the parameters in (4.1) as discussed in connection with Table 4.1, we are gratified that the statistical <:f240,2Times New Roman,0,0,0> accuracy of the results from our simulation is already sufficient<:f><:f240,2Times New Roman,0,0,0> to resolve the difference between the <:f240,2Symbol,0,0,0>a<:f240,2Times New Roman,0,0,0> and <:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0> phases. <:f><:f240,2Times New Roman,0,0,0>We therefore claim that we have successfully (a) predicted (and we mean 'predicted', before the experimental<:f><:f240,2Times New Roman,0,0,0> determination) the Oh occupation ratio in <:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<:f240,2Times New Roman,0,0,0><+'>3<-'><:f><:f240,2Times New Roman,0,0,0> and (b) demonstrated the effect of oxygen framework topolo gy on the Al site distribution, by using<:f><:f240,2Times New Roman,0,0,0> our very simple model.<:f><-!> <:#280,9360> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#280,9360> <:s><:#280,9360> <:s><:#280,9360> <:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#376,9360><:f320,,><+!>VII.6 Discussion :<-!><:f><+!><:f320,,> Geometrical Interpretation of Results<-!><:f> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#2024,9360><:f240,2Times New Roman,0,0,0>The purpose of this section is to explore how far the structural properties of<:f240,2Times New Roman,0,0,0> <:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<+'>3<-'><:f240,2Times New Roman,0,0,0> <:f>can be accounted for in terms of the geometrical fcc packing of oxygen atoms in <:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<+'>3<-'><:f240,2Times New Roman,0,0,0>. In particular, we ask whether<:f><:f240,2Times New Roman,0,0,0> there are some inherent features of the hcp and fcc oxygen lattices that account for th e strong ordering on Oh sites only in one case (<:f240,2Symbol,0,0,0>a<:f>) and occupation of both Oh and T sites in the other (<:f240,2Symbol,0,0,0>g<:f>). In the latter case, can the presence of occupied T sites destroys the long range order ? We shall se e that we can indeed interpret the results of the simulations, and experimentally observed reality, along such structural lines. <:s><:#280,9360> <:#2520,9360>We shall consider the oxygen structure built up of close-packed layers, stacked ABCABC for the fcc oxygen structure and ABABAB for the hcp one, in the usual notation. The Al atoms lie on the Oh and T sites between pairs of oxygen layers (Fig.6.1). We know f rom the simulations of Section 4 that the Oh sites have lower energy and we therefore start by considering the Al atoms on Oh sites only. We assume that each Al layer to have the same density of Al because it is energetically favourable. The arrangement of Al atoms, occupying two-thirds of the Oh sites because of the <:f240,2Times New Roman,0,0,0>Al<+'>2<-'>O<+'>3<-'><:f240,2Times New Roman,0,0,0> composition, is unique and rather obvious (Fig.6.1). It is best reviewed<:f><:f240,2Times New Roman,0,0,0> by not ing that the empty Oh sites are spread out as uniformly as possible in a triangular mesh<:f><:f240,2Times New Roman,0,0,0> (shown as + in <:f>Fig.6.1). We will refer to Oh and T sites between A and B oxygen layers as Oh<:f240,2Times New Roman,0,0,0><+'>(AB<:f240,2Times New Roman,0,0,0><-'><+'>)<-'><:f> and T<+'>(AB)<-'>. <:s><:#280,9360> <+B><:#280,9360><[>Fig.6.1] <+B><:#240,9360><:f200,,>Arrangement <:f><:f200,,>of Al atoms (large shaded circles) on Oh sites between a pair of oxygen <:f><:f200,,>closed-packed <+B><:#240,9360><:f200,,>layers.<:f><:f200,,> The oxygens of the upper layer are at the vertices of the triangular <:f><:f200,,>lattice <:f><:f200,,>and<:f><:f200,,> those<:f><:f200,,> of lower<:f> <+B><:#240,9360><:f200,,>layer at the small dots (only shown on the left).<:f><:f200,,> The hexagons <:f><:f200,,>on the let are the projection of three <+B><:#240,9360><:f200,,>octahedra, tow filled with Al and on empty.<:f><:f200,,> The <:f><:f200,,>empty<:f><:f200,,> Oh sites are marked on the right side of the <+B><:#240,9360><:f200,,>figure by +. Two types<:f><:f200,,> of (empty) T<:f><:f200,,> sites <:f><:f200,,>lie directly above the lower (A) layer and directly <:f><:f200,,>below <+B><:#240,9360><:f200,,>those of upper (B) layer. <:f240,2Times New Roman,0,0,0><:A7> <:#280,9360><:f240,2Times New Roman,0,0,0> <:#280,9360> <:#7080,9360>We now turn to the three-dimensional stacking of oxygen and Al layers, starting with the hcp oxygen structure. The Oh(BA') sites between BA' pair of oxygen layers are directly above the Oh<:f240,2Times New Roman,0,0,0><+'>(AB)<-'><:f> sites of previous (AB) pair of oxygen layers in the ABA'B'A''B'' stacking of the hcp oxygen structure, where the primes simply distinguish successive layers of the same type. Let us assume that the Al atoms between the A and B oxygen layers conform to the arrangement of Fig.6.1. The Oh<:f240,2Times New Roman,0,0,0><+'>(BA')<-'><:f> sites of the next BA' pair of oxygen layers lie directly above the Oh<+'>(AB)<-'> sites and we can arrange that the Al atoms on Oh<:f240,2Times New Roman,0,0,0><+'>(BA')<-'><:f> sites follow the same pa ttern but displaced by the vector <+!>t<-!> shown in Fig.6.1. In this way all the sites above the empty Oh<:f240,2Times New Roman,0,0,0><+'>(AB)<-'><:f> sites (shown as + in Fig.6.1) are filled which must be energetically most favourable because this minimi ses the number of face-sharing Al octahedra between two layers. That accounts for the half of the Al atoms between the BA' oxygen layers, and the other half then spread themselves out to make the unique pattern of Fig.6.1 again but displaced by <+!>t<-!> as already mentioned. We conclude that for ABABAB oxygen packing, a very good, low energy structure results from placing <+">all<-"> Al in Oh sites, with those in one Al layer displaced by <+!>t <-!>with respect to the adjacent one : this is not true for the fcc ABCABC stacking as we shall see below. We now turn to adding the third Al layer, assuming (without lost of generality) that the second Al layer is displaced by +<+!>t<-!> with respect to the first. The third layer can be displaced by +<:f240,2Times New Roman,0,0,0><+!>t<-!><:f240,2Times New Roman,0,0,0> <:f>with respect to the second, yielding Fig.6.2.(a), or <:f240,2Symbol,0,0,0>-<+!><:f>t<-!> yielding Fig.6.2.(b). These are clearly not equivalent. The former has lower energy because the Al atoms are more uniformly spread out, with no second neighbour fac e-sharing octahedra. In Fig.6.2.(a), half of the Al atoms are in continuous columns as shown by the triple circles. We conclude that we would expect the lowest energy structure based on hcp oxygen packing to have Al layers displaced successively by +<+!>t,<-!> +<+!>t,<-!> +<+!>t,<-!> +<+!>t<-!>, ..., and this is indeed the <:f240,2Symbol,0,0,0>a<:f>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f>O<+'>3<-'> structure. We note that each Al has only <+">one<-"> face-sharing neighbour across the oxygen layers, i.e. one-half such pairing per atom. The dominant energy cost of the stacking is therefore (1/2)<+">J<-">(r<+'>face<-'>) per atom if we take into account the decrease of J(r<+'>ij<-'>) with distan ce, which is very low compared with the fcc result as we shall see below. <:s><:#280,9360> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <+B><:#280,9360><:f240,2Times New Roman,0,0,0><[>Fig.6.2] <+B><:#240,9360><:f200,2Times New Roman,0,0,0>Al atoms (large, medium and small circles<:f><:f200,2Times New Roman,0,0,0>) in Oh sites between successive pairs AB, BA' and <:f><:f200,2Times New Roman,0,0,0>A'B' <+B><:#240,9360><:f200,2Times New Roman,0,0,0>of oxygen layers in an fcc oxygen structure, where A', B' are next A-type and B-type layers. <+B><:s><:#280,9360><:f240,2Times New Roman,0,0,0> <+B><:#280,9360><:f240,2Times New Roman,0,0,0><[>Fig.6.2.(a)] <+B><:#248,9360><:f200,2Times New Roman,0,0,0>As in <:f200,2Symbol,0,0,0>a<:f200,2Times New Roman,0,0,0>-Al<:f200,2Times New Roman,0,0,0><+'>2<-'><:f200,2Times New Roman,0,0,0>O<+'>3<-'><:f><:f> <+B><:f240,2Times New Roman,0,0,0><:A6> <+B><:s><:#280,9360><:f240,2Times New Roman,0,0,0> <+B><:#280,9360><:f240,2Times New Roman,0,0,0><[>Fig.6.2.(b)] <+B><:#240,9360><:f200,2Times New Roman,0,0,0>In an unobserved structure <+B><:s><:f240,2Times New Roman,0,0,0><:A5> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:f240,2Times New Roman,0,0,0>We can now see what happens when we try to apply the same types of argument to <:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<+'>3<-'> with ABCABC oxygen stacking. We start again with the Al atoms as in Fig.6.1 in the first Oh<+'>(AB)<-'><:f><:f240,2Times New Roman,0,0,0> layer. The next oxygen layer is now C-type so that the Oh<+'>(BC)<-'> sites are now differently placed from the Oh<+'>(BA')<-'> sites before, as shown in Fig.6.3 : i.e. <+">adjacent<-"><:f><:f240,2Times New Roman,0,0,0> layers of Oh sites have a different relation in the fcc oxygen lattice from that in the hcp case. All the Oh<+'>(BC)<-'> sites are equivalent<:f><:f240,2Times New Roman,0,0,0> in Fig.6.3 as regards their po sitioning with respect to the Al atoms in Oh<+'>(AB)<-'>, so that the lowest energy structure will again be that of the hexagonal rings in Fig.6.1, now displaced by one of the three equivalent<:f><:f240,2Times New Roman,0,0,0> vectors pointing to site 1, 2 and 3 indicated in Fig.6.3. Fig.6.4 shows one such arrangement, and we can assess its energy in a similar way to before. Each Al atom in Oh<+'>(BC)<-'> now has two edge-sharing contacts with Al in Oh<+'>(AB)<-'> octahedra and we can write the dominant term in the stacking energy as 2<+">J<-">(r<+'>edge<-'>) per atom. We have r<+'>edge<-'> = 1 d<+'>O-O<-'> (d<+'>O-O<-'> is the ideal oxygen close-pack distance) and r<:f240,2Times New Roman,0,0,0><+'>face<-'><:f240,2Times New Roman,0,0,0> = <:A0>d<:f240,2Times New Roman,0,0,0><+'>O-O<-'><:f> so that r<:f240,2Times New Roman,0,0,0><+'>edge<-'><:f>/r<:f240,2Times New Roman,0,0,0><+'>face<-'><:f><-'><:f240,2Times New Roman,0,0,0> = 1.22. Thus we expect 2<+">J<-">(r<:f240,2Times New Roman,0,0,0><+'> edge<-'><:f240,2Times New Roman,0,0,0>) to be substantially<:f><:f240,2Times New Roman,0,0,0> larger then (1/2)<:f240,2Times New Roman,0,0,0><+">J<-"><:f240,2Times New Roman,0,0,0>(r<:f240,2Times New Roman,0,0,0><+'>face<-'><:f240,2Times New Roman,0,0,0>) a nd thus Al inter-layer energy in fcc oxygen structure to be substantially<:f><:f240,2Times New Roman,0,0,0> higher then in the hcp case, i.e., we expect that the <:f240,2Symbol,0,0,0>a<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<+'>3<-'> structure will have lower energy than <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<+'>3<-'> based on an fcc oxygen structure.<:f> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <+B><:#280,9360><:f240,2Times New Roman,0,0,0><[>Fig.6.3] <+B><:#240,9360><:f200,2Times New Roman,0,0,0>Al atoms (shaded circles) in Oh sites between one AB pair of close-packed oxygen and all Oh <+B><:s><:#240,9360><:f200,2Times New Roman,0,0,0>sites (marked "Oh") between the next BC pair of oxygen layers in an fcc oxygen structure. <:s><:f240,2Times New Roman,0,0,0><:A4> <:#280,9360><:f240,2Times New Roman,0,0,0> <+B><:#280,9360><:f240,2Times New Roman,0,0,0><[>Fig.6.4] <+B><:#240,9360><:f200,2Times New Roman,0,0,0>Al atoms (shaded circles) in Oh sites between an AB pair of oxygen layers, and one of <+B><:#240,9360><:f200,2Times New Roman,0,0,0>three equivalent<:f><:f200,2Times New Roman,0,0,0> arrangements of Al atoms (empty circles) in OH sites between the next <+B><:#240,9360><:f200,2Times New Roman,0,0,0>BC <:f><:f200,2Times New Roman,0,0,0>pair of oxygen layers in an ABCABC fcc oxygen structure. <:s><:f240,2Times New Roman,0,0,0><:A3> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#3376,9360>We can now consider whether the energy in the fcc oxygen framework can be lowered further by moving some Al atoms from Oh to T sites. The optimum configuration of two adjacent Al layers based on Oh sites along has already been derived in Fig.6.4 and is repr oduced in Fig.6.5.(a). A typical Al atom such as number 6 is edge sharing with 5 other Al octahedra, namely those numbered 1, 5 and 10 plus two Al in Oh<+'>(AB)<-'> shown as shaded circles. If it is displaced by vector shown in Fig.6.5.(a) to a neighbouring T site several of these bonds are broken, and if Al atom 5 move simultaneously out of the way by the vector also shown in Fig.6.5.(a), we obtain the structure shown in Fig.6.5.(b). This has three less edge-sharing Al neighbours but there is now one T-site Al which cost an energy |<:f240,2Symbol,0,0,0>m<:f>| in the sense of equation (1.1) and (4.1). The tetrahedron occupied by atom 6 only shares corners with surrounding Al octahedra, so that its electrostatic energy is not high. We conclude that the movement of some Al atoms from Oh to T sites such as in Fig.6.5.(a),(b) is expected to lower the total energy, as shown by the simulation in Section 5. <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <+B><:#280,9360><:f240,2Times New Roman,0,0,0><[>Fig.6.5]<:f> <+B><:s><:#280,9360><:f240,2Times New Roman,0,0,0> <+B><:s><:#280,9360><:f240,2Times New Roman,0,0,0><[>(a)] <+B><:#240,9360><:f200,2Times New Roman,0,0,0>Two Al atoms, numbered 5 and 6, taken from Oh sites in the configuration <+B><:#240,9360><:f200,2Times New Roman,0,0,0>of <:f><:f200,2Times New Roman,0,0,0>Fig.6.4 on <:f><:f200,2Times New Roman,0,0,0>an fcc oxygen structure, and moved to tetrahedral (T) sites. <:s><:f240,2Times New Roman,0,0,0><:A2> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <+B><:#280,9360><:f240,2Times New Roman,0,0,0><[>(b)]<:f> <+B><:#240,9360><:f200,2Times New Roman,0,0,0>The final configuration <+B><:s><:f240,2Times New Roman,0,0,0><:A1> <+B><:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#3640,9360><:f240,2Times New Roman,0,0,0>The final step in our argument is to consider<:f><:f240,2Times New Roman,0,0,0> why the occupation of T sites may destroy<:f><:f240,2Times New Roman,0,0,0> the long range order in the structure based on the fcc oxygen lattice. There are of course many pairs of Al atoms such as 2,3 and 8,9 and 7,11 which could make a similar transformation to Fig.6.5.(a),(b) resulting in a T site. However once atoms 5 and 6 have transformed, it is no longer energetically favourable for the 7,11 pa ir to do so too because the movement of atom 5 has already removed an edge-sharing neighbour from atom 7. Similarly<:f><:f240,2Times New Roman,0,0,0>, the equivalent<:f><:f240,2Times New Roman,0,0,0> transformation for the pair 8,9 in Fig.6.5.(b) would plac e atom 8 too close to atom 6, making this more energetically unfavourable. Such arguments apply altogether to four pairs around the 5,6 pair. Thus the total number of occupied T sites is not expected to be very high, with an exclusion zone without other Al T-atoms around each one. The process of generating Al-T-atoms can therefore nucleate in different places in mutually incompatible<:f><:f240,2Times New Roman,0,0,0> ways, thus destroying the long range order. An equivalent way of expressing the same thing is to say that a modest proportion of Al on T-sites generates a substantial configurational entropy, which can destroy long range order at a relatively low temperature<:f><:f240,2Times New Roman,0,0,0>. <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#1728,9360><:f240,2Times New Roman,0,0,0>In conclusion, we have seen that we can make plausible arguments to explain what is observed in the simulations and experimentally on <:f><:f240,2Symbol,0,0,0>a<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<+'>3<-'> <:f><:f240,2Times New Roman,0,0,0>and <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<+'>3<-'>, namely (a) that the Al ordering in <:f><:f240,2Times New Roman,0,0,0> <:f240,2Symbol,0,0,0>a<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<+'>3<-'> <:f><:f240,2Times New Roman,0,0,0>is very good leading to its being th lowest energy structure, in comparison<:f><:f240,2Times New Roman,0,0,0> with other Al ordering patterns on Oh sites in an hcp oxygen lattice, (b) a similar good Al ordering pattern cann ot be achieved on the Oh sites only for an fcc lattice, leading to some Al on T-sites in <:f><:f240,2Symbol,0,0,0>a<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<+'>3<-'>, and (c) the occupation of some T sites in <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<+'>3<-'> tends to destroy the long range order.<:f> <:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360> <:s><:#280,9360> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#280,9360><:f240,2Times New Roman,0,0,0> <:#376,9360><+!><:f320,,>VII.7 Plans for Possible Further<-!><:f><+!><:f320,,> Work<-!><:f> <:s><:#280,9360> <:#280,9360> <:#1432,9360><:f240,2Times New Roman,0,0,0>The research<:f><:f240,2Times New Roman,0,0,0> on <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<+'>3<-'> was stopped at a convenient<:f><:f240,2Times New Roman,0,0,0> point after completing what is really an extensive preli minary survey<:f><:f240,2Times New Roman,0,0,0>. The gratifying surprise is that it has already yielded an understanding of the main feature of <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<+'>3<-'>, both semi-qualitatively<:f><:f240,2Times New Roman,0,0,0> in terms of the simulations of Section 5 and the geometrica l insight in Section 6. This fully justifies the basic methodology<:f><:f240,2Times New Roman,0,0,0> of the using of the model Hamiltonian as an intermediate step.<:f> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#1696,9360><:f240,2Times New Roman,0,0,0>However some modest tidying up is clearly indicated. the simulation of a and g structures need repeating as proper simulated anneals at temperature T with the Metropolis<:f><:f240,2Times New Roman,0,0,0> algorithm. Then the res ults need more detailed analysis of the short and long range order. A further avenue of research<:f><:f240,2Times New Roman,0,0,0> is to improve the fit of the model Hamiltonain to the 9 energies calculated <+">ab initio<-">, by doing a least square fit or perhaps by expressing the <+">J<-">(r) as a Coulomb term plus local strain corrections for nearest neighbour<:f><:f240,2Times New Roman,0,0,0> face, edge or corner sharing of occupied octahrdra and tetrahedra. <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#848,9360><:f240,2Times New Roman,0,0,0>Much more extensive computing would<:f><:f240,2Times New Roman,0,0,0> be required either to extend significantly the data base of <+">ab initio<-"> configurations, or what<:f><:f240,2Times New Roman,0,0,0> is really more important, to relax the oxygen positions and thus take local stain effect into account. <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#2848,9360><:f240,2Times New Roman,0,0,0>However one of the original hope has been to address surface properties, including perhaps the energies of different crystal<:f><:f240,2Times New Roman,0,0,0> surface orientations. These may throw light on the reason why <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<+'>3<-'> is almost only found as a very fine microcrystalline powder. In view of what has been said, an extensive <+">ab initio<-"> <:f><:f240,2Times New Roman,0,0,0>study of surface energy of different surface is (regarding the power of computers today) out of the question. However we can speculate and propose one or two likely structures based on our model Hamiltonia n<:f><:f240,2Times New Roman,0,0,0> and perhaps do <+">ab initio<-"> calculations of total energy relaxing oxygen and Al positions but not site occupancies. A 3-step approach similar to the one introduced in this chapter, with extra fit-able<:f><:f240,2Times New Roman,0,0,0> terms in model Hamiltonian<:f><:f240,2Times New Roman,0,0,0> that account for the surface configuration, can also be used to study the structure and energetics of the <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<+'>3<-'> surfaces.<:f><:f240,2Times New Roman,0,0,0><-"> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#3128,9360><:f240,2Times New Roman,0,0,0>One way to rationalise the<:f><:f240,2Times New Roman,0,0,0> fact that <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<+'>3<-'> <:f><:f240,2Times New Roman,0,0,0>almost only exist in fine powder is to say that nucleation energy<:f><:f240,2Times New Roman,0,0,0> is very low because <:f><:f240,2Times New Roman,0,0,0>the surface energy is very low. The point is that with <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<+'>3<-'> <:f><:f240,2Times New Roman,0,0,0>being disordered, the Al configuration at a surface can itself to minimise the surface energy which might result in a very low energy value, at least for some specific surface orientation(s). This could be explained within the existing model Hamiltonian<:f><:f240,2Times New Roman,0,0,0> and perhaps followed by one or two <+">ab initio<-"> calculations mentioned. However one note of caution should be mentioned. The model Hamiltonian<:f><:f240,2Times New Roman,0,0,0> does not contain the long range Coulomb interaction between Al ions. This may not be very important for the bu lk structure with Al more or less uniformly<:f><:f240,2Times New Roman,0,0,0> spread out, but could be an important aspect of surface energies. This could probably be modelled by a shell model simulation as an intermidiancy between our full <+">ab initio<-"> technique and the rather crude model Hamiltonian. <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#328,9360><:f280,2Times New Roman,0,0,0><+!> <:#328,9360><+!><:f280,2Times New Roman,0,0,0> <:#328,9360><+!><:f280,2Times New Roman,0,0,0> <:#328,9360><+!><:f280,2Times New Roman,0,0,0>Acknowledgments<-!><:f> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0>I would like to thank : <:#280,9360><:f240,2Times New Roman,0,0,0>Mr. C.-F. Cheng's close collaboration<:f><:f240,2Times New Roman,0,0,0> with us on the experimental<:f><:f240,2Times New Roman,0,0,0> investigation of the material. <:#296,9360><:f240,2Times New Roman,0,0,0>Dr. X. Yuang for kindly provide us part of the <:f240,2Symbol,0,0,0>a<:f240,2Times New Roman,0,0,0> and <:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<+'>3<-'> sample. <:#296,9360><:f240,2Times New Roman,0,0,0>Dr. Macrod bring <:f><:f240,2Symbol,0,0,0>g<:f240,2Times New Roman,0,0,0>-Al<+'>2<-'>O<+'>3<-'><:f><:f240,2Times New Roman,0,0,0> to our interest and collaborated with us at the early<:f><:f240,2Times New Roman,0,0,0> stage of the project. <:#296,9360><:f240,2Times New Roman,0,0,0>Dr. S. Bhattacharjee for the discussion on <:f240,2Symbol,0,0,0>a<:f240,2Times New Roman,0,0,0>-Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f240,2Times New Roman,0,0,0>O<:f240,2Times New Roman,0,0,0><+'>3<-'><:f> at the early stage of the work. <:#280,9360><:f240,2Times New Roman,0,0,0>Prof. McConnell for his inspiring suggestion on the projects of alumina in general. <:#280,9360><:f240,2Times New Roman,0,0,0>Prof. Andaus Carson for the discussion on cluster-cluster interaction. <:#280,9360><:f240,2Times New Roman,0,0,0>Prof. C. Cheng for useful discussion of the error tolerance total energy in a variety<:f><:f240,2Times New Roman,0,0,0> of systems. <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#328,9360><:f280,2Times New Roman,0,0,0><+!>References<-!><:f><:ZReferences><:Z~References> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0><[>C.1] <:#288,9360><:f240,2Times New Roman,0,0,0>Y.T. Chu, J.B. Bates, C.W. White , and G.C. Farlow, J. Appl. Phys. <+!>64<-!> (7) 3727-3730 (1988) <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0><[>C.2] <:#288,9360><:f240,2Times New Roman,0,0,0>T.C. Chou and T.G. Nieh, J. Am. Ceram. Soc., <+!>74<-!> <[>9] 2270-2279 (1991) <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#280,9360><:f240,2Times New Roman,0,0,0><[>C.3] <:#568,9360><:f240,2Times New Roman,0,0,0>C.R. Catlow and W.C. Mackrodt (eds), Computer Simulation of Solid, Lecture Note in Physics, <+!>166<-!> (1982) <:#280,9360><:f240,2Times New Roman,0,0,0> <:#280,9360><:f240,2Times New Roman,0,0,0><[>D.1] <:#288,9360><:f240,2Times New Roman,0,0,0>B.G. Dick and A.W. Overhauser, Phys. Rev. <+!>112<-!>, 90 (1958) <:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0><[>G.1] <:#280,9360><:f240,2Times New Roman,0,0,0>N.N. Greenwood and A. Earnshaw, Chemistry of The Elements, Ch.7, Pergamon Press (1990) <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#280,9360><:f240,2Times New Roman,0,0,0><[>H.1] <:#280,9360><:f240,2Times New Roman,0,0,0>V. Heine, private communications by W.C. Mackrodt, D. Bird, P. Madden and others <:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0><[>I.1] <:#288,9360><:f240,2Times New Roman,0,0,0>H. Iizuka, K. Yokoo and S. Ono, Appl. Phys. Lett. <+!>61<-!> 2975 (1992) <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0><[>O.1] <:#280,9360><:f240,2Times New Roman,0,0,0>H. O'Neill, A talk given in "ESF Progam on Kinetics of Minerals and Ceramics" (4th Feb 1994)<:f> <:#280,9360><:f240,2Times New Roman,0,0,0> <:#280,9360><:f240,2Times New Roman,0,0,0><[>P.1] <:#568,9360><:f240,2Times New Roman,0,0,0>M.C. Payne, M.P. Teter, D.C. Allan, T.A. Arias and J.D. Joannopoulos, Rev. Mod. Phys. <+!>64<-!> No.4, 1045 (1992) <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0><[>R.1] <:#288,9360><:f240,2Times New Roman,0,0,0>A. Reller and D.L. Cocke, Catalysis Letters <+!>2<-!> (1989) 91-96 <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0><[>S.1] <:#288,9360><:f240,2Times New Roman,0,0,0>J. Schafer and C. J. Adkins, J. Phys.: Condens. Matter <+!>3<-!> 2907-2915 (1991) <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0><[>S.2] <:#280,9360><:f240,2Times New Roman,0,0,0>R. Serimaa, Acta Polytechnica, Applied Physics Series No. 169 (1990) <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0><[>S.3] <:#568,9360><:f240,2Times New Roman,0,0,0>P.S. Sklad, J.C. McCallum, C.J. McHargue and C.W. White, Nuclear Instruments and Method in Physics Research <+!>B<-!>46 102-106 (1990) <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0><[>W.1] <:#280,9360><:f240,2Times New Roman,0,0,0>S.J. Wilson and J.D.C. Mc Connell, J. Solid State Chem, 34, 315-322 (1980)<:f> <:s><:#280,9360><:f240,,0,0,0> <:s><:#280,9360><:f> <:#280,9360><:f240,2Times New Roman,0,0,0> > Times New Roman,18,12,0,0,0,0,0 $\sqrt{(3/2)}$SMTSuS hF`aF`aFFSS`(`(F 5 F 5 9 O 9 O S" "S `  ` ,R R ,` `  S S _`  _ ` `   ` S~  ~S S  S   SS    SH}HS}  H` H` {S  {S d dUdU du z u z  u^ u @ z z^ z @ u u^ dW \ W \  WG W   " \ \G \ "   W WG dZ_Z_ ZZ%___U%ZUZ d05}05} 00i555E}}i}0E0 d`e`e ` `+ee eR+`R`  d     T   =   T   =   T dpXuP pXuP ppXX;Xuuu;PPPpp d !   !ibi! b  dch" ch cc.hhh_.c_c d #!  > <>    <  > d  $"     :      b  :  b  dm Xr P%#m Xr P m m  X X8 Xr r r 8 P P Pm m  dd\&$d\ d.dxd$x\.\\$ d/ 4 '%/ 4  /T / h   4 4T 4   h / /T d (&  > Y>  Y    > dW O)'W O >WWW   OO>O d JB*( JB  BJJJ BBBB    dL Q +)L Q  L ? L    Q Q ? Q  L L ? d  ,*     / y    _y /   _  d-+ b_b_ d .,  K  h K   h   K d}  /-}   }b }   H  b  H   } }b d 0.  T  ( p  T p ( T d1/8 :3:3: d208 2C2|C|2 d G 31 G 8  x /`   G /G xG   `   x d&42&8 %?&&%&o?o% d/X53/X8 /</qXX<Xq//< d8a648a8 898zaa9az889 d  75  8  N 7   N     7   d  86  8  N  D  N    D    dQ  z  97Q  z  8 Q Q N   8  z N z z 8    Q Q d :8 8  j !q!jq  j d - ;9 - 8  A F   -A - -   F   de <:e 8 e}e4L4}L   ee} d  =;  8  5  " u   5 u "    5 d~><~8 ~ ~e We~W~  dHq?=Hq8 H H/qq qW/HWH  d@>8 m$-$m-m d  A?  8    m     W m   W  d B@ 8    E WE  W  d8CA88 >Q88>8Q> d I DB I 8  ~ 5bI5I~I  b  ~ d% EC% 8 %%  > 6>%6% JFDxTms Rmn1 JPGExTms Rmn2 J|(HF|xTms Rmn3 JuIGxTms Rmn4 J ~ JH xTms Rmn8 JS  KIS xTms Rmn9 JS LJSxTms Rmn7 KMKTms Rmn100 JMZNLMx Tms Rmn5 JMOMMx Tms Rmn6 dC l PNC l 8 C D C * l l D l * C C D d   QO   8   @ [   @    [     [  RP Tms RmnAl in Oh(AB) sites d  SQ  8  D ! s   D  s ! D ]  TR Tms RmnAB)Al bewteen BC layers K   S Tms Rmn @11SMUTS. ?F`aTF`aFSUFSRTS`(QS`(F 5 PRF 5 9 O OQ9 O S" NP"S ` MO ` ,R LNR ,` KM`  S JL S _` IK_ ` ` HJ ` S~ GI ~S S FH S  EG SDFS  CE  SH}BDHS} AC H` @BH` {S ?A {S d>@ dUdU d  =?   ^   R   ^   R  ^ dW \ <>W \  WG W   " \ \G \ "   W WG dZ_;=Z_ ZZ%___U%ZUZ d05}:<05} 00i555E}}i}0E0 d`e9;`e ` `+ee eR+`R`  d  8:   T   =   T   =   T dpXuP79pXuP ppXX;Xuuu;PPPpp d 68   !ibi! b  dch57ch cc.hhh_.c_c d 46  > <>    <  > d  35     :      b  :  b  dm Xr P24m Xr P m m  X X8 Xr r r 8 P P Pm m  dd\13d\ d!dkd$k\!\\$ d/ 4 02/ 4  /T / h   4 4T 4   h / /T d /1  > Y>  Y    > dW O.0W O >WWW   OO>O d JB-/ JB  BJJJ BBBB    dL Q ,.L Q  L ? L    Q Q ? Q  L L ? d  +-     / y    _y /   _  d*, b_b_ d )+  K  h K   h   K d}  (*}   }b }   H  b  H   } }b d ')  T  ( p  T p ( T \ 5 &( 3Tms RmnAl in Oh(AB) sites d%'8 `K`K` d$&8 2C2|C|2 d G #% G 8  x /`   G /G xG   `   x d"$8 alala d   !#   8   @ [   @    [     d/X "/X8 /</qXX<Xq//< d8a!8a8 898zaa9az889 d     8  N 7   N     7   d    8  N  D  N    D    dQ  z  Q  z  8 Q Q N   8  z N z z 8    Q Q d  8  j !q!jq  j d -  - 8  A F   -A - -   F   de e 8 e}e4L4}L   ee} d    8  5  " u   5 u "    5 d~~8 ~ ~e We~W~  dHqHq8 H H/qq qW/HWH  d8 m$-$m-m d    8    m     W m   W  d  8  C s C s C d888 >Q88>8Q> d I  I 8  ~ 5bI5I~I  b  ~ d% % 8 %%  > 6>%6% JxTms Rmn1 JPxTms Rmn2 J|( |xTms Rmn3 Ju xTms Rmn4 J ~  xTms Rmn8 JS  S xTms Rmn9 JS SxTms Rmn7 K Tms Rmn100 J_ x Tms RmnF^_MM]M U WV5& Jx Tms RmnF^_MM]M U WV6&r8r8(0JJ(0 d  8    E WE  W  [ .   Tms RmnAB Al in Oh(BC) sitesB K    Tms Rmn 5  " u   11SMHGS F`aF`aFFSS`(`(F 5 F 5 9 O 9 O S" "S `  ` ,R R ,` `  S S _`  _ ` `   ` S~  ~S S  S   SS    SH}HS}  H` H` {S  {S d dUdU d     ^   R   ^   R  ^ dW \ W \  WG W   " \ \G \ "   W WG dZ_Z_ ZZ%___U%ZUZ d05}05} 00i555E}}i}0E0 d`e`e ` `+ee eR+`R`  d     T   =   T   =   T dpXuP pXuP ppXX;Xuuu;PPPpp d !   !ibi! b  dch" ch cc.hhh_.c_c d #!  > <>    <  > d  $"     :      b  :  b  dm Xr P%#m Xr P m m  X X8 Xr r r 8 P P Pm m  dd\&$d\ d!dkd$k\!\\$ d/ 4 '%/ 4  /T / h   4 4T 4   h / /T d (&  > Y>  Y    > dW O)'W O >WWW   OO>O d JB*( JB  BJJJ BBBB    dL Q +)L Q  L ? L    Q Q ? Q  L L ? d  ,*     / y    _y /   _  d-+ b_b_ d .,  K  h K   h   K d}  /-}   }b }   H  b  H   } }b d 0.  T  ( p  T p ( T [V P 1/V Tms RmnAl in Oh(AB) sites d208 IGIGI da31a8 a^aH^Haa^ d G 42 G 8  x /`   G /G xG   `   x d & 53 & 8  H ? & & H & ? H d0 640 8 6I0600  I  d758 hNhNh dIr 86Ir 8 II60r6rr0   II d  97  8  N 7   N     7   d" - :8" - 8  ^ " S" " ^   - S- -   d3 ! \ , ;93 ! \ , 8 3 3 ] u ! !  ! \ ] \ \  , , u , 3 3 d <: 8  ^ d^d  ^ d ! 2, =; ! 2, 8   ] K! ! ! 2] 2 2 , , K,   d   ><   8   8 X   8    X      dAj?=Aj8 APA(jjPj(AAP dR{@>R{8 R R9{{ {W9RWR  d+TA?+T8 + +mTT TWm+W+  d7`B@7`8 767y``6`y776 d  CA  8    m     W m   W  d DB 8    E WE  W  d)REC)R8 )D)kRRDRk))D dT}FDT}8 TrT);})}r};TTr d  GE  8  B  7    B  7    B dvHFv8 vrv) ])r] vvr [  G  Tms RmnAB Al in Oh(BC) sitesrSMTSS F`aF`aFFSS`(`(F 5 F 5 9 O 9 O S" "S `  ` ,R R ,` `  S S _`  _ ` `   ` S~  ~S S  S   SS    SH}HS}  H` H` {S  {S d dUdU d     ^   R   ^   R  ^ dW \ W \  WG W   " \ \G \ "   W WG dZ_Z_ ZZ%___U%ZUZ d05}05} 00i555E}}i}0E0 d`e`e ` `+ee eR+`R`  d     T   =   T   =   T dpXuP pXuP ppXX;Xuuu;PPPpp d !   !i Pi! P  dch" ch cc.hhh_.c_c d #!  > <>    <  > d  $"     :      b  :  b  dm Xr P%#m Xr P m m  X X8 Xr r r 8 P P Pm m  dd\&$d\ d!dkd$k\!\\$ d/ 4 '%/ 4  /T / h   4 4T 4   h / /T d (&  > Y>  Y    > dW O)'W O >WWW   OO>O d JB*( JB  BJJJ BBBB    dL Q +)L Q  L ? L    Q Q ? Q  L L ? d  ,*     / y    _y /   _  d-+ b_b_ d .,  K  h K   h   K d}  /-}   }b }   H  b  H   } }b d 0.  T  ( p  T p ( T K[{1/[{(Tms Rmnuu;PPPp Oh K:_bo20:_(Tms Rmnuu;PPPp Oh K31(Tms Rmnuu;PPPp Oh K42(Tms Rmnuu;PPPp Oh K53(Tms Rmnuu;PPPp Oh K ! 64 (Tms Rmnuu;PPPp Oh K  75 (Tms Rmnuu;PPPp Oh K(P 86((Tms Rmnuu;PPPp Oh K 97(Tms Rmnuu;PPPp Oh K*:8(Tms Rmnuu;PPPp Oh K{ ;9{(Tms Rmnuu;PPPp Oh K  <: (Tms Rmnuu;PPPp Oh K {=; {(Tms Rmnuu;PPPp Oh K{><{(Tms Rmnuu;PPPp Oh K5]?=5(Tms Rmnuu;PPPp Oh K  @> (Tms Rmnuu;PPPp Oh K%A?(Tms Rmnuu;PPPp Oh K3[ B@3(Tms Rmnuu;PPPp Oh K , CA (Tms Rmnuu;PPPp Oh KDB(Tms Rmnuu;PPPp Oh K<EC(Tms Rmnuu;PPPp Oh K  FD (Tms Rmnuu;PPPp Oh K  GE (Tms Rmnuu;PPPp Oh Kt HFt(Tms Rmnuu;PPPp Oh K5] IG5(Tms Rmnuu;PPPp Oh Km  JHm (Tms Rmnuu;PPPp Oh K= e  KI= (Tms Rmnuu;PPPp Oh K  4 LJ  (Tms Rmnuu;PPPp Oh K  MK (Tms Rmnuu;PPPp Oh Uq  NLq Tms RmnAl in Oh(AB) di  OMi  Tms Rmnuu;PPPp Oh : Oh sites in Oh(BC)r8PNr8(0 %MQOM %(0abRPab(0 J[hSQ[x Tms Rmn1= JbTRx Tms Rmn2= JZ  SZ x Tms Rmn3=SMlkv ,"F`aF`aFFSSF 5 F 5 9 O 9 O c c | | S" "S ` ` Sh hS,R  R ,`  `  S  S _` _ ` `  ` S~  ~S S  S   SS    UUSH}HS}  H` H` {b  {b dz666z6z dC\ HT C\ HT C C |\ \ \ H H H T T |T C C dh /` h /`   Xh h h / / /( ` ` X` (  d "\"g\g" d\ a !\ a \D \ ' a aD a ' \ \D dW \ " W \ WG W   " \ \G \ "   W WG dZ_#!Z_ZZ%___U%ZUZ d05}$"05}00i555E}}i}0E0 d`e%#`e``+eeed+`d` d  &$  f "  =  " f   =   f dpXuP'%pXuPppXX;Xuuu;PPPpp d (&   !ibi! b  dch)'chcc.hhh_.c_c d *( > <>    <  > d  +)    :      b  :  b  dy j~ b,*y j~ by y  j jD j~ ~ ~ *D b b by *y  dd\-+d\d!dkd$k\!\\$ d/ 4 .,/ 4 /T / h   4 4T 4   h / /T d /- > Y>  Y    > dh `0.h `>hhh   (``>`( d JB1/ JB  BJJJ BBBB    d5 : 205 : 5 K 5  n   :  : K : n 5 5 K d  31     / y    _y /   _  d42b_b_ d 53 ]  h ]   h   ] d}  64}  }b }   H  b  H   } }b d 75  T  ( p  T p ( T t86tPQV97PQV dXX:8XY(XXX=XgXXXXgX=XXXX d[;9[- stack [/[/`[`[/ d-0<:-0- stack --S 00S0- - da  8=;a  8- stack a a        8 8 8a a  d6{(><6{(- stack 66\{{{((\(66 d Y <?= Y <- stack    3 Y Y Y 3 < < <   d \ @> \ - stack C    6 \ \C \v 6   v C d6aA?6a- stack  66 6;aaa;  dqB@q- stack 9iKqiq9qK dxCAx- stack !;k!Rxkx;xR! dq[DBq[- stack Hqqq5H[[[5 d-EC-- stack $--$-WW$ d  FD  - stack h 8  B s 8 h  s B   h dk GEk - stack D  Ek kD ku E   u D d= HF= - stack 5   =  = 5 = f    f 5 d4 IG4 - stack 4D 4 Z D u   Z 4u 4D d JH - stack K  *[ K | [ *  | K d' KI' - stack ' ` ' 0 M } 0 ` } M ' ' ` d  qLJ  q- stack   3 d    Kd q3 q q K  dSZMKSZ- stack SSy4ZZyZS4S d 3 NL 3 - stack K     3 3K 3|    | K d v OM v - stack =   P v v =v nP    n = d PN - stack D   A r  D w r A  w D d7QO7- stack 'W7W7'7 ddRPd C2C//MdddM/ dRSQR C2CRRiiRR d,  TR,   C2Ca C -, K, i, C a  i K -  a dp  USp   C2CpU p7    7 U s    ps pU dx9VTx9 C2Cxx"999x"x dXWUX C2CXXooXX d T XV T  C2CJ ,   = T, TJ Th =   h J dfYWf C2C1OfffO1 d > ZX >  C2C    $ > > > $      d o P[Y o P C2C   : X o o o 9X P: P P 9  dI ) \ZI )  C2CI ^ I @ ` ) ~ ) ) @ ^ | ~ ` I | I ^ d<W][<W C2C<"<Sq"@WqWSW<@<" dX s ^\X s  C2CX> X o    > \ s s os X\ X> d @ _] @ - stack Z )    @) @Z @     Z d  `^   C2CK -    - K i    i K U  a_ ; Tms RmnF 5 < ? Al in Oh(AB) V~ b`~ HTms RmnTms RmnF 5 Al in Oh(BA') W v ca pTms RmnF 5  iAl in Oh(A'B')x dBdbB C2C  +BBB+  d5ec5 C2C555 dT^fdT^ C2C) =T T)TG=^^^G) dY  ^geY  ^ C2CY )Y p     ) G ^ ^p ^Y GY ) d:Xhf:X C2C:#:Qo#AXoXQX:A:# d'Xig'X C2C#''#'AXXXA# d?/  jh?/   C2C?d ?F V/ t/ / F d   t V ? ?d d   ki    C2C J  ,    , J h     h  J d)  lj)   C2C^ @ ) ) ) @ ^ |    | ^ d7 C k7 C  C2Cl N 7 7 ,7 CN Cl C ,    l SMih !F`aF`aFFSS`(`(F 5 F 5 9 O 9 O c c | | S" "S ` ` Sh hS,R  R ,`  `  S  S _` _ ` `  ` S~  ~S S  S   SS    UUSH}HS}  H` H` {b  {b dz666z6z dC\ HT C\ HT C C |\ \ \ H H H T T |T C C dh /` h /`   Xh h h / / /( ` ` X` (  d "\"g\g" d\ a !\ a \D \ ' a aD a ' \ \D dW \ " W \ WG W   " \ \G \ "   W WG dZ_#!Z_Z Z%__ _O%ZOZ  d05}$"05}00i555E}}i}0E0 d`e%#`e``+eeed+`d` d  &$  f "  =  " f   =   f dpXuP'%pXuPppXX;Xuuu;PPPpp d (&   !ibi! b  dch)'chcc.hhh_.c_c d *( > <>    <  > d  +)    :      b  :  b  dm Xr P,*m Xr Pm m  X X8 Xr r r 8 P P Pm m  dd\-+d\d!dkd$k\!\\$ d/ 4 .,/ 4 /T / h   4 4T 4   h / /T d /- > Y>  Y    > dh `0.h `>hhh   (``>`( d JB1/ JB  BJJJ BBBB    d5 : 205 : 5 K 5  n   :  : K : n 5 5 K d  31     / y    _y /   _  d42b_b_ d 53 ]  h ]   h   ] d}  64}  }b }   H  b  H   } }b d 75  T  ( p  T p ( T t86tPQV97PQV dXX:8XY(XXX=XgXXXXgX=XXXX d[;9[- stack [/[/`[`[/ d4?<:4?- stack 44Z??Z?44 d 8 8=; 8 8- stack     8 8 8  8 8 8   d6u"><6u"- stack 66\uuu""\"66 d G 0?= G 0- stack    ! G G G ! 0 0 0   d @ @> @ - stack Z )    @) @Z @     Z d  A?  - stack j :  F w : j  w F   j dOqB@Oq- stack OOuKqquqOKO dxCAx- stack !;k!Rxkx;xR! dq[DBq[- stack Hqqq5H[[[5 d9yEC9y- stack 999QyyyQ d  FD  - stack h 8  6 g 8 h  g 6   h dk GEk - stack D  Ek kD ku E   u D d= HF= - stack 5   =  = 5 = f    f 5 d4 IG4 - stack 4D 4 Z D u   Z 4u 4D d JH - stack K  *[ K | [ *  | K d  KI  - stack  Y  ) >  n   ) Y n >   Y d  qLJ  q- stack   3 d    Kd q3 q q K  dSZMKSZ- stack SSy4ZZyZS4S d 3 NL 3 - stack K     3 3K 3|    | K d'OM' C2C''>\zz\>'' d v PN v - stack =   P v v =v nP    n = d QO - stack D   A r  D w r A  w D d7RP7- stack 'W7W7'7 d\SQ\ C2C 'E\\\E'  d  TR   C2C              d1US1 C2C111 dM\VTM\ C2C' 6M M'ME6\\\E' dmPWUmP C2Cmm9PPPm9m dR  VXVR  V C2CR !R i     ! ? V Vi VR ?R ! dK?YWK? C2C 4KK K(4???(  d=  ZX=   C2Ck P = = = P k      k d4Y[Y4Y C2C4$4Ki$BYiYKY4B4$ d S p\Z S p C2C ;   < S S ;S Y< p p p Y ; d*Y][*Y C2C***>YYY> d;-  ^\;-   C2C;b ;D R- p- - D b   p R ; ;b d  n _]  n  C2C9         9  W  n n n W 9 d  ~ `^  ~  C2C I +    + I g ~ ~ ~ g I d \ a_ \  C2CN 0  ' E \0 \N \l E '  l N d&  b`&   C2C[ = & & & = [ y    y [ dy#  cay#   C2CyX y: # # # : X v    yv yX d  w ~ db  w ~  C2C I + $  B  `  w + w I w g ` ~ B ~ $ ~ g I d2 : ec2 :  C2Cg I 2 2 #2 :I :g : #    g d6gfd6g- stack 666AgggA d{cge{c C2C{.{.Lccc{L{. U  hf ; Tms Rmn:XoAl in Oh(AB)e V  ig HTms Rmn:X iAl in Oh(BA')d W$  h$ pTms Rmn:X iAl in Oh(A'B')SMi F`aF`aFFSS`(`(F 5 F 5 9 O 9 O c c | | S" "S ` ` Sh hS,R  R ,`  `  S  S _` _ ` `  ` S~  ~S S  S   SS    UUSH}HS}  H` H` {b  {b dz666z6z dC\ HT C\ HT C C |\ \ \ H H H T T |T C C dh /` h /`   Xh h h / / /( ` ` X` (  dZ_ Z_ZZ%___U%ZUZ d ! > <>    <  > d  "     / y    _y /   _  d#!b_b_ \  $" Tms RmnD~~~BzAl (all in Oh site);%#;t&$tk'%kck(&kcV.)'V.^*(^'m+)'mz,*z}k-+}kk.,k/-wu0.uwPQV1/PQVj|^20j^|;31;I;J42IJ;]|^53]|^ J*- 64`@Times New Roman?/JY #YF$$Ft dXX75XY(XXX=XgXXXXgX=XXXX Z  86  Tms RmnOxygen of layer AF`a97F`aF:8FS;9S`(<:`(F 5 =;F 5 9 O ><9 O c ?=c | @>| S" A?"S ` B@ ` ShCAhS,R DBR ,` EC`  S FD S _` GE_ ` ` HF ` S~ IG ~S S JH S  KI SLJS  MK  UNLUSH}OMHS} PN H` QOH` mH RP mH dSQz666z6z dC\ HT TRC\ HT C C |\ \ \ H H H T T |T C C dh /` USh /`   Xh h h / / /( ` ` X` (  dldVTld l+lul,ud+dd, d  WU   7  A    7 |   A | 7 dd i XVd i  d9 d   / i i9 i~ /   d~ d9 dZ_YWZ_ ZZ%___U%ZUZ d=B|ZX=B| ==vBBBD||v|=D= d`e[Y`e ``+eee_+`_` d  \Z   T   0 z  T  z 0   T dbYgQ][bYgQ bbYY-Yggg-QQQbb d ^\   /wTw/ T  dIN_]IN IINNN_I_I d `^  > <>    <  > d  a_     :      U  :  U  d e ]b` e ]   e eR e   %R ] ] ] %  dca g[g[ d" ' db" '  "F " [   ' 'F '   [ " "F d ec  K  Y K  Y    K dd\fdd\ 1dydd$\y\1\$ d IAge IA  BIII AABA    d5 : hf5 :  5 ? 5 n   : : ? : n 5 5 ? d  ig     / y    _y /   _  djh b_b_ d ki  L  h L   h   L d}  lj}   }G }   H  G  H   } }G d mk  G  ( p  G p ( G \  nl Tms RmnD~~~BzAl (all in Oh site);om;tpntkqokckrpkcV.sqV.^tr^'mus'mzvtz}kwu}kkxvkywwuzxuwPQV{yPQVj|^|zj^|;}{;QBR~|QRB]|^}]|^G H ~G H <0 dXXXY(XXX=XgXXXXgX=XXXX d 0 0(      000   d(((((( db!b!(bbp!!p!bb dkQkQ(+/DYkk+k@YQDQ/Q@+ dfYfY(f9f(u(9KYYuYfKf9 d9Jw9Jw(9e9VFJWJiJwVwewuiWF9u9e dN+N+(n\N NN+\+n+ n d#g#g(D2###2DWgggWD dQ Q (m]QQQ ] m }}m dV|V|(VVf|y||yfVV dee(reeer d B r B r ( Y L B B B L Y g r r r g Y Z   Tms RmnOxygen of layer Add..H 0 I H 0 I           $  $  t  t2 4 3 2 4 3 "j#"j#()j()j DJ E DJ E ^ h d ^ h d Times New Roman,18,12,0,0,0,0,0 $$E=E_0^{\prime }+\mu \cdot \,\text{N}_{\text{Oh}}\ +\underset{\text{ }}\to{\overset{\text{6}}\to{\underset{\text{m=1}}\to{\sum }}}\,J(\text{r}_{\text{m}})\cdot n(\text{r}_{\text{m}})\ $$SS;xx` [Embedded] 14 .tex 130260 47 0 0 9 .sdw 130307 6504 0 0 10 .sdw 136811 6465 0 0 11 .sdw 143276 5586 0 0 12 .sdw 148862 5779 0 0 13 .sdw 154641 8748 0 0 7 .sdw 163389 8448 0 0 8 .sdw 171837 8329 0 0 3 .tex 180166 220 180386 16774 00197162