What is the method for calculating ion radii in a NaCl structure?

[broegger]

Hi,

I have yet another problem with this diffraction thing. I have little clue on these questions:

e) The radius of the O^2- ion is assumed to be 0.126 nm. By x-ray diffraction experiments the dimensions of the unit cells for MgO, CaO, SrO and BaO has been determined to be 0.4213 nm, 0.4811 nm, 0.5160 nm and 0.5539 nm. All four compounds has the NaCl structure. Calculate the radii of the cations: Mg^2+, Ca^2+, Sr^2+ and Ba^2+.

f) ZnSe has a cubic close packed structure of the anions, but with the cations in the tetrahedral holes. How large a fraction of the tetrahedral holes are occupied in this structure?

Any hints?

 

[mezarashi]

X-ray diffraction is a useful tool in probing the structure of materials. Here we already have the length dimensions derived for you, so less physics, more material science.

NaCl is an example of an FCC (face centered cubic) structure. If you draw a picture of the cross-section, you see that along the diagonal, you have the radius of the oxygen, followed by the diameter of the cation, and then once again the radius of the oxygen. Since FCC is cubic, the diagonal direction is square root two the lattice constant. Can you draw a relationship between the radius and lattice constants now?

As for the tetahedral holes, they are interstitials rather than part of the body structure due to the huge mismatch of atomic radii. This is a bit of test on your geometry. If you put four atoms together in 3D space, what is the radius of the atom that can fit inbetween the 'hole space' between them. Start it easy using 2D. Like for example take 3 nickels and put them together so the sides touch. See how there is space in between. What is the radius of the circle that could fit in there? There is then a definite relationship between radius and volume.

 

[broegger]

Thanks, man. But how do I know that the cations and anions exactly touch (no free space between them), which is the assumption in your calculation (right?). Aren't they, like, hovering with some free space between them? (this is probably a stupid question :-)

 

[mezarashi]

Hmmm, yes, we are assuming they perfectly touch. I don't think it's a 'stupid' question. It's a very good one. Firstly, did you know that atoms are 99.99% empty space? The nucleus is a dot with an electron cloud covering it. We can define the atomic radius by the distance in which another atom can approach it without 'penetrating' one anothers electron cloud. All of this "hovering" thus has been included in the atomic radii given in the tables ^^

 

[broegger]

I see. Thanks for taking the time! This is off-topic, but how can we even talk about free space, when electrons are clouds that, theoretically, penetrate all space with some small amplitude.

And also, a totally unrelated question: What is the scope of the Pauli exclusion principle. It states that two particles in a system can't be in the same state - if the "system" is an atom it obviously applies, but if the "system" is the whole universe (or the Earth or whatever) it does not (or does it??); I mean, electrons in Denmark does not affect the state of electrons in Singapore, right? Maybe I should start a new thread...

 

[mezarashi]

Your raising a lot of 'deep' issues. I would say the quantum physicists here would be better answering your questions. I'll address a few nonetheless.

The electron cloud does NOT penetrate "all space". The electroncs 'orbit' the atom at certain discrete energy levels. These energy levels relate to the distance from the atomic nucleus. So they cannot just jump to any distance they want, unless they have the energy. Even if they do, most of the time they will quickly lose it through radiation and cool back down to their original state. There is a certain distance range in which they achieve equilibrium from a classicial and quantum point of view.

About the Pauli exclusion principle. I think you are confusing this with entanglement (particles affecting each other from a distance). The exclusion principle disallows atoms and electrons to have the same quantum state, and they don't. That's why we have electron shells and different energy bands. They all want to achieve a low potential state, but only so many can go to each state, so some need to have more energy than others to not violate this. Note as well that NOT ALL particles obey the exclusion principle. Photons for example do not. This is why we have Bose-Einstein statistics in addition to Fermi-Dirac.

 

[broegger]

About the Pauli exclusion principle. I think you are confusing this with entanglement (particles affecting each other from a distance). The exclusion principle disallows atoms and electrons to have the same quantum state, and they don't. That's why we have electron shells and different energy bands. They all want to achieve a low potential state, but only so many can go to each state, so some need to have more energy than others to not violate this. Note as well that NOT ALL particles obey the exclusion principle. Photons for example do not. This is why we have Bose-Einstein statistics in addition to Fermi-Dirac.

Ok, but two electrons in two different atoms can have the same set of quantum numbers without violating the exclusion principle. Why is this not a violation and when does it become a violation? (i.e. how close must the atoms be.) I hope you understand my question.