chemia - Studia
chemia - Studia
chemia - Studia
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OMEGA POLYNOMIAL IN P-TYPE SURFACE NETWORKS<br />
The polynomials are calculated on a cubic lattice of dimension (k,k,k), at<br />
R max [8]; following similar considerations and analyzing the calculations<br />
made by our original Nano Studio [16] software, we derived the formulas,<br />
listed in Table2, and provided examples for some k-values, as well.<br />
Figure 2. The Z_P (left) and A_P (right) crystal-like structures<br />
Table 1. Topological data for the units of Z_P and A_P structures<br />
Octahedral Vertices Edges Faces Open Omega Polynomial CI<br />
structure<br />
f 8 Faces<br />
R max [8]<br />
Z_P 120 168 12 6<br />
1 2 4<br />
48X + 36X<br />
+ 12X<br />
27840<br />
A_P 144 192 12 6<br />
2 4 6<br />
12X + 24X + 12X 36000<br />
In the Z_P and A_P network structures, the term at exponent 8<br />
represent the number of edge strips of length 8; these strips cross only f 8<br />
when link 4 Z_P units, and cross faces f 8 and f 6 when link 4 A_P units<br />
respectively, so it is present starting with k=2. In case of Z_P net, the term<br />
at exponent 8 counts the large hollows, ordered as in zeolites, natural<br />
alumino-silicates, used as molecular sieves or in chemical catalysis.<br />
Table 2. Omega polynomial in Z_P and A_P networks<br />
Formulas for Z_P network<br />
2 1 2 2 2 4 2 8<br />
Ω ( X, k, Rmax[8]) = 48k X + 12 k(4k − 2k+ 1) X + 3 k(5k + 3k− 4) X + 3 k( k−1)<br />
X<br />
'<br />
2<br />
2<br />
2<br />
2 2<br />
Ω (1) = 48k<br />
+ 2 ⋅12k(4k<br />
− 2k<br />
+ 1) + 4 ⋅3k(5K<br />
+ 3k<br />
− 4) + 8⋅3k(<br />
k −1)<br />
= 12k<br />
(15k<br />
−1)<br />
''<br />
2<br />
Ω (1) = 12k<br />
(37k<br />
− 23k<br />
+ 4)<br />
5 4 3 2<br />
CI ( k)<br />
= 48k(675k<br />
−90k<br />
+ 3k<br />
−13k<br />
+ 6k<br />
−1)<br />
Formulas for A_P network<br />
Ω(<br />
X , k,<br />
R<br />
+ k<br />
2<br />
4<br />
2 8 2 5k<br />
1<br />
max[8])<br />
= 12kX<br />
+ 12k(<br />
k + 1) X + 3k(<br />
k −1)<br />
X + 12k<br />
X + 24k∑<br />
i=<br />
2<br />
2<br />
X<br />
4(2i−1)<br />
'<br />
Ω (1) = 12k<br />
(15k<br />
+ 1)<br />
''<br />
3 2<br />
Ω (1) = 4k<br />
(203k<br />
+ 33k<br />
− 80k<br />
+ 12)<br />
5<br />
4<br />
3 2<br />
CI ( k)<br />
= 4k(8100k<br />
+ 1080k<br />
−167k<br />
− 78k<br />
+ 77k<br />
−12)<br />
k Omega polynomial: examples CI<br />
R max [8], Z_P network<br />
1 48X+36X 2 +12X 4 27840<br />
2 192X+312X 2 +132X 4 +6X 8 1933728<br />
3 432X+1116X 2 +450X 4 +36X 8 22567104<br />
4 768X+2736X 2 +1056X 4 +108X 8 128288064<br />
5 1200X+5460X 2 +2040X 4 +240X 8 492768960<br />
6 1728X+9576X 2 +3492X 4 +450X 8 1478124000<br />
R max [8] A_P network<br />
213
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