What Determines the Void Space in Cubic Crystalline Structures?

[Reshma]

The problem here is to find the maximum radius of the interstitial sphere that could just fit into the void space of cubic crystalline structures of:
the simple cube
body centered cubic
face centered cubic

My question is how is position of the void space determined in these structures?

 

[inha]

Use the hard sphere model ie. there are spheres of radius r at the lattice points and their radius and the lattice constant are connected via a relation for each of the unit cells.

 

[Reshma]

Use the hard sphere model ie. there are spheres of radius r at the lattice points and their radius and the lattice constant are connected via a relation for each of the unit cells.

Thank you for replying.
I'm aware of relationship between the lattice constant and the atomic radius.
Let 'a' be the lattice constant and 'r' be the atomic radius 'r'.
For simple cubic structure: $$r = \frac{a}{2}$$

For body centered:$$r = a\frac{\sqrt{3}}{4}$$

For face centered: $$r = a\frac{\sqrt{2}}{4}$$

But how is the position of the void determined exactly(using geometry)?

 

[inha]

Pick a lattice plane and draw the spheres for visual aid. It shouldn't be difficult to see where you can fit an interstitial atom. Then draw another sphere of unknown radius r' and solve for it in the same manner the a-r relations are normally solved.

 

[PrinceOfDarkness]

In simple cubic's case, it is quite simple.

For BCC: Consider the body diagonal. It's length is aXsqrt3 (Pythagorus Theorem!).

For FCC: Consider the face diagonal. It's length is aXsqrt2 (Pythagorus again!).

 

[armandowww]

And now: what about diamond bonding angle?

 

[inha]

Uh, what about it?

 

[Gokul43201]

My question is how is position of the void space determined in these structures?

By intuition ?

I can't think of any simple scheme that will tell you where the largest interstitials are...but if you spend a little time (or a lot, depending on how familiar you are with crystal geometries) picturing the structure, you can easily guess where these positions are. For the SC, it's pretty obvious where the biggest void is. This gets a little harder for the BCC and the FCC, but in both cases, a little clever thinking will get you home.

If not, have you come across octahedral and tetrahedral voids ? You might want to give these a look...