Baddeleyite has the space group symmetry P21/c with four ZrO2 in the unit cell. It has a unit cell dimensions of:
a = 5.169 b = 5.232 c = 5.341 Å (all ± 0.008 Å), β = 99˚15ˊ ± 10ˊ.
The coordination number for ZrO2 has been found to be 7.
How can I estimate the number of 'formula units' in one unit cell of monoclinic ZrO2 (crystal structure)?
Not sure if I get the question, in the second picture you see the four Zr-ions in a unit cell and also write in your text it has 4 formula units. where is the problem?
I want to know the number of atoms per unit cell of monoclinc ZrO2.
First image was Zr coordination polyhedron in baddeleyite. So Can I take that as unit cell for monoclinic ZrO2?
No, you can't use the first one in a straightforward way but the second is appropriate. 4 * 3 gives you 12 atoms per unit cell. Simple, right?
Ok. then How many numbers of atom in (111) plane of monoclinic ZrO2? And what is most densely packed plane for the same?
I answered your other question about the surface, when counting atoms make sure to only count the corner atoms as 1/4 and border atoms as 1/2 since they're only partially in the cell, in VESTA you can also reduce any cell to its primitive version by using utilities => niggli-reduced cell.
From the operational point of view, Monoclinic ZrO2 structure could be obtained by uniform shear transformation along the one of those principal axis of fluorite structure (Ca F2), plus the some extensions along the principal axis (a,b,c). In a naive way, Ca F2 is isomorphic to FCC structure having all those tetrahedral interstitial sites (8) are occupied by F and the substitutional sites of FFC (4) are occupied Ca. Even though according to the report, if one goes from the high temperature fluoride structure to the room temperature monoclinic lattice, the coordination of Zr4+ changes from 8 to 7 in, still total number of atoms per unit cell seems to me 4+8= 12.(See: WELLS)
Another rule in crystallography is that the number of occupied lattice sites obeying the Group symmetry is even number, the multiple of 2 (2, 4, 6, 8, ......).
Note: Coordination number of Zr4+ was reported by J.D. McCullough and K.N. Trueblood Acta Cryst. 1959, V12, p.507).
There are 12 number of atoms per unit cell of Monoclinic ZrO2 because of Z=4.
The number Z of formula units per unit cell can be calculated using the formula
Z = DVN/M
where: D = specific gravity (gr/cm*3); V = volume of unit cell (cm*3, a.b.c.sin(beta) in you case); N = Avogadro number (6.022.10**23 formula units/mole; M = weight of the unit formula (gr/mole) = sum of the atomic weights of the atoms included in the unit formula (2 O and 1 Zr in your case). In your case Z = 4, therefore you have 8 O and 4 Zr atoms in a unit cell. See the following site http://en.wikipedia.org/wiki/Baddeleyite .
One easy way to know the no.of atoms in unit cell is looking at its Pearson symbol. Pearson Symbol for Monoclinic ZrO2 is mP12, which means its a Primitive monoclinic structure with 12 no.of atoms in unit cell.
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