# Crystal structures

1. Nov 6, 2008

Hey All..

I'm having some problems trying to understand the tetrahedral and octahedral sites in face centered cubic (fcc) and body centered cubic (bcc). It is hard to visualise the structures in 3D..

I wanted to know how to find out the positions per unit cell for the tetrahedral and octahedral in both fcc and bcc..

For the tetraheral
fcc = 8 tetrahedron per unit cell
bcc = 12 tetrahedron per unit cell

The bcc is worked out (6*4)/2 = 12

For the octrahedral
fcc = 4 octrahedron per unit cell
bcc = 6 octrahedron per unit cell

The fcc is worked out (12/4)+1 = 4
The bcc is worked out (12/4)+(6/2) = 6

Although I have the soultions I'm still confused how they are worked out to the above answers..I have attached a picture.. hope someone can help

Thank you

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2. Nov 6, 2008

### Gokul43201

Staff Emeritus
First, count the number of sites (small open circles, in the attached pictures) of each kind. In the FCC there are two kinds of oct-sites, marked in red and white. Do you understand why they are different?

Next, you look at each kind and determine whether or not the site is entirely contained within the unit cell. In the above example, only one kind of site is. Which one?

Whenever a site is shared by more than one unit cell, you need to figure out how many unit cells it is shared between. The white oct-sites on the FCC live on the unit cell edges. How many unit cells share each such edge? Divide the number of shared sites of each type by the number of unit cells that share it*, to get the number per unit cell.

* Note: This division assumes that the site is shared equally by each of the unit cells involved in the sharing. This is often the case (determine by inspection from symmetry), but need not always be.

Last edited: Nov 6, 2008
3. Nov 7, 2008

My understanding of the topic is very brief so bare with me..

Firstly I'm not sure about the difference between the red and white..
The FCC e.g. you have given is an octahedral site

From my understanding of per unit cell for fcc is 4 and bcc is 2.. the way i know this is by counting the solid spheres as in the attached pic..

Thank you

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4. Nov 7, 2008

### Gokul43201

Staff Emeritus
That is the number of lattice sites per unit cell. You need to perform a similar counting for the number of interstitial sites per unit cell. And there are two kinds of such interstitial sites: octahedral and tetrahederal.

Do you know what an interstitial site (or void) is?

That's where you need to start before you can count them.

5. Nov 7, 2008

defects in the materials..?

6. Nov 7, 2008

maybe the question should be how do you find the number of voids for tetrahedral/octahedral around sphere...?

(but then why are there 12 white circles in the fcc...and not 6
whereas the tetrahdral there are 8..)

arghh i'm confusing myself!!

not sure if i'm making sense sorry

7. Nov 7, 2008

### Gokul43201

Staff Emeritus
No, the interstitials we are talking about here are not defects - they are the spaces between the atoms that occupy lattice sites. The geometry of atoms surrounding the void determines whether you call it a tetrahedral void or an octahedral void.

You seem to be pretty confused. You need to start from the beginning here. Do you have a textbook?

8. Nov 8, 2008

Thanks I found the website useful..

No I don't have a textbook just some notes..but I need to understand the basics before I move on to different structures rock salt, zinc blende etc..

I have also come across this a few times in internet searches (from the link)..
"the particular sphere being considered covers a triangular void in the layer above it and another in the layer below it. Thus two more tetrahedral voids surround the spheres. This results in 2 * 3 + 1 + 1 = 8 tetrahedral voids and 2 * 3 = 6 octahedral voids surrounding the sphere. Since a tetrahedral void is shared by four spheres, there are twice as many tetrahedral voids as there are spheres. Similarly, since an octahedral void is surrounded by six spheres, there are as many octahedral voids as there are spheres"

maybe a picture of this may help me understand it better..

Last edited: Nov 8, 2008
9. Nov 8, 2008

Thanks now I understand what you meant :).. I wasn't dividing! silly mistake .. and rather counting all the circles.. thanks again for that..

but now I'm still confused on the 2*3=6 octahedral voids and 2*3+1+1=8 tetrahedral voids.. lol sorry

Last edited: Nov 8, 2008
10. Nov 13, 2008

How to you determine no. of octahedral/tetrahedral voids around sphere..
(2*3=6 oct, 2*3+1+1=8 tet)

What is the best way to visualise this?

11. Nov 13, 2008

Hey All..

I found the answer I worked it out... found a useful site so thought I will share :)
Very good for crystal structures.. exactly what I needed to help visualise the stuff..

The answer to my question above.. Voids around sphere for tetrahedral/octahedral
I have attached a picture..

#### Attached Files:

• ###### phf_oct-tet_voids_around_sphere.gif
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12. Nov 13, 2008

### Gokul43201

Staff Emeritus
Sorry, I forgot to get back to this thread. Glad you found what you needed.

13. Nov 15, 2008