From XRD data, you can get the unit cell parameters. Using the unit cell parameters, a,b,c, you can get the unit cell volume. Based on the crystal structure of the material, you can find how many molecular units are present in every unit cell. You will also know the molecular weight of the material as you know the chemical structure. Using all these data and using the formula, and the formula outlined, you should be able to calculate the theoretical density.

Theoretical density = (Molecular weight x No. of molecules per unit cell) / (Volume of unit cell x Avogardo's number)

You will have to find the X-ray data of this system. Once you know the crystal structure, you can calculate the electron concentration (i.e. average electronic number density) with the aid of the periodic table of elements. For details, see Chapter 1 of the Solid State book by Ashcroft & Mermin.

I'm not sure if I get why you need the electron concentration, but it is certainly true that the true density of you mixed oxide needs XRD data. I would say that the value obtained from XRD the way you show it in your pdf-file is exactly what you are searching for, however, provided that the concentration of yttria is known precisely enough and is appropriately accounted for by the software (in other words, make sure that the program uses not the molar weight of pure zirconia, but a value adjusted for partial substitution of Zr-ions with Y-ions). Hope I could explain it clear enough. Good luck!

You should Rietveld refine your XRD/NPD data to get accurate unit cell volume, then identify different types of atoms in the unit cell and multiply their numbers/fractions with the respective atomic masses to get the formula mass in the unit cell, and then use the formula that you have given in the pdf attachment.

In fact, the theoretical density can't be reached, with the many defects in the sample. Assuming that the crystal is totally perfect and only the unit cell periodically repeats in it. Then the theoretical value can be calculated with the cell volume and the cell mass.

Mass/volume of the unit cell can give you theoretical density of the material.

As written before the density is very easy to calculate by the quotient between unit cell mass and unit cell volume. Both is standard in XRD simulation software based on single crystal structure data so that you can usually find in all of these programmes the "X-Ray density". Whereas the cell volume is practically error-free (during calculation) the correct generation of the unit cell mass depends on a correct crystal structure generation. For your system practically irrelevant are missing structure components like crystal water or OH groups which are not always described by a poistion in crystal structure reports and data bases. They are often only mentioned as "+2 H2O molecules). During structure generation the mass of these molecules are then not determined and also neigher considered in the cell mass nor in the density. This is essential since then the scaling factors in quantitative phase analysis are incorrect. Even for a few hydrogen atoms this can cause remarkable errors if these non-described structure components are not corrected in another way.

The density itself is actually only important for the derivation of atom% from weigth%, e.g. in software packages. If we could count the unit cells for ech phase in a multicomponent powder we wouldn't need the density. But it is easier to weight different phase and mix them instead of a producing a mixtur of 20at% of element A and 80at% of element B.

Lattice (or structure) defects like dislocations and vacancies will decrease the densities whereas interstitials or a radiation with neutrons or protons should increase the density of a phase. In so far I would relativise the statement of Xiaoye Liu, although in the majority of cases he is probably right.

Excellent answers from the material science point of view, now you know how it is done, but look at the peculiarities of yttria stabilized zirconia. Have a look at the phase diagram first (e.g. Chen, Hallstedt, Gauckler).

At ambient temperature "tetragonal" Y-TZP with typically 2.5 -3.5 mol-% yttria is is a non equilibrium material. And its crystal structure can shift from a solid solution of yttria in zirconia which is either homogeneously distributed (supersaturated and non-equilibrium) at low sintering temperature or more or less segregated towards the boundaries of the t+F field which is a miscibility gap. Then you have cubic and tetragonal grains. So the material you want to measure as a reference is not clearly defined.

We found differences between 6.05-6.12 g/cm³ in hot pressed fully dense (acc. SEM) Y-TZP of 3 mol-% of stabilizer depending on stabilizer distribution and sintering parameters.

The cubic materials with >13 mol-% yttria are a different story here the methods shown above are fully valid.

The answer to this question can be read my article under the title "Thermal analysis of friction welding process in relation to the welding of YSZ-alumina composite and 6061 aluminum alloy" . The Article published on the Applied Surface Science Journal. You can look Table 1 (Bulk density and theoretical density of the ceramic compositions investigated). The article attached with this comment.

Article Thermal analysis of friction welding process in relation to ...

From XRD data, you can get the unit cell parameters. Using the unit cell parameters, a,b,c, you can get the unit cell volume. Based on the crystal structure of the material, you can find how many molecular units are present in every unit cell. You will also know the molecular weight of the material as you know the chemical structure. Using all these data and using the formula, and the formula outlined, you should be able to calculate the theoretical density.

Theoretical density = (Molecular weight x No. of molecules per unit cell) / (Volume of unit cell x Avogardo's number)

Dear Poulami,

Can we take No. of atoms per unit cell in place of molecules.

 Dear Poulami Dutta,

Thank you for your share, I really need to calculate unit cell parameters a, b,c, V and material density from XRD data. I am working with Barium ferrite and I also like to know how to compute percentage of Barium having in samples. Please help me, thanks a lot.

Check this Link,

http://www.diracdelta.co.uk/science/source/t/h/theoretical%20density/source.html#.WJ89l99KEwg 

Dear Beddiaf Zaidi 

Thank you for your share the Link

The attached paper of title: 

CALCULATION OF DENSITY FROM X-RAY DATA

will help you.

Regards

how to calculate density when we have variable density unit cells in our lattice structure?

There is no such thing as theoretical density of a condensed phase (liquid or solid). There is no way a priori to know how the atoms/molecules will pack together in a crystal lattice in the solid state, nor how closely they will approach each other in the liquid state. If you are told the type of crystal (e.g., face-centered cubic) and the dimensions of the unit cell, then you could calculate a theoretical density. But you didn’t mention that, so I will assume you didn’t mean that.

A “theoretical density” of sorts can be calculated for elements in the gas phase by assuming they are ideal gases. The ideal gas equation can be rearranged to give:

ρ=m/V =Pm0/RT

where m0m0 is the molar mass of the gas atom or molecule.

Density is an emerging quantity and can not be calculated (theoretically) from the atoms themselves. Densities as we know them, e.g. in condensed matter, have indeed exclusively been measured, e.g. by diffraction experiments, When you know the structure of a system, you know the distance between its atoms and you are able to calculate the particle density. Know, in addition, the involved atoms and you can also calculate the mass density.

But, the formation of structures and densities is a natural, self-organizing process. By the coupling betwen many atoms, by their occupation of equilibrium position, atoms get a size and their structural arrangement defines a density. Size of an atom and densities of many atoms are therefor emerging quantities. More precise: by cooperations with others, each attending atom gets a size, but dependent also on the neighbors: ionic radius, covalent radius, ..). Subsequently, a size of an atom is not the size of an atom by itself - instead, depends also on the neighbors.

Subsequently, measure the distances and you know the size of the involved atoms in a given surrounding. Calculate the sice of the atoms, account the atoms´ structural arrangement, and you get something you may call a particle density. From this you can subtract the mass density. A clear way to succeede, but it´s an experimental result.

Theoretical density? That an extremely different task. Atom by atom, would be an exhausting task, in general not well understood and, in particular, a noway to anywhere. Quantum mechanics just forget - too many atoms/particles involved. No hope to solve ever. Assume 100 Al-atoms (just a few and no other atoms of different type), 3 electrons each (2s+1p) and the core (four particles for each atom), just ignore its inner electrons, which may have to be proven. Still a boring and completely un-interesting system. Imagine how many parameters you need to describe each particle 4 x the number of further paraameters of each internal particle (position, momentum, charge, etc.), multiply by 100 and you end up with a few thousand parameters - now start to solve Schroedinger - let me know which hospital you are staying!

Sometimes, with some ( a lot) assumptions, putting in things you may know from other experiments of other quantities, by trying different ways of data-handling, by squinting at least a bit to the final result you may already know in advance, by playing a lot with all the parameters, and you get somethin, a result. Beautiful! Publish immediately, but only when you got what you knew in advance (from where else then an experiment) - otherwise you loose your reputation.

Cooperative behaviour will be a solution! It may reduce the number of parameters a lot. You may skip Schroedinger (for sure ohne her). .... There are some techniques, well under development. Similar techniques alreaady exist in other fields and they work perfect, we all know.

I'm very sorry, its very too long (was even longer in the mean time). But, its an interesting question and not solved yet in all details - Sorry again, I just took a break.

When we obtain the dimensions of the unit cell and the number of atoms in it from XRD of the sample , we can apply the following equation Dth(g/cm3) =1.66N M/ V(Ao) , otherwise, we cannot obtain the theoretical density, which represents the density of unit cell and is thus equal to the exact density of the material without pores or any crystal defects.