[Big-Daddy]

In the unit cell, silicon atoms are placed with their centres at each corner of the cube, and in the centre of each face. If we divide the unit cell into eight smaller cubes, as shown on the right, there is also one silicon atom right in the middle of every alternate cube. The unit cell is a cube of length a pm.

Derive an expression for the Si-Si bond length in terms of a.

If it's a face centred cubic, Face Diagonal = (root 2)·Edge Diagonal. The face diagonal contains 2 Si-Si bonds so Si-Si Bond Length = (1/2)·Face Diagonal=(1/2)·(root 2)·Edge Diagonal. But this is apparently wrong. Where's my mistake?

[curiouscat]

In the unit cell, silicon atoms are placed with their centres at each corner of the cube, and in the centre of each face. If we divide the unit cell into eight smaller cubes, as shown on the right, there is also one silicon atom right in the middle of every alternate cube. The unit cell is a cube of length a pm.

Derive an expression for the Si-Si bond length in terms of a.

If it's a face centred cubic, Face Diagonal = (root 2)·Edge Diagonal. The face diagonal contains 2 Si-Si bonds so Si-Si Bond Length = (1/2)·Face Diagonal=(1/2)·(root 2)·Edge Diagonal. But this is apparently wrong. Where's my mistake?

There is no right.

[curiouscat]

Your mistake: You ignored this:

If we divide the unit cell into eight smaller cubes, as shown on the right, there is also one silicon atom right in the middle of every alternate cube.

[Big-Daddy]

Your mistake: You ignored this:

If we divide the unit cell into eight smaller cubes, as shown on the right, there is also one silicon atom right in the middle of every alternate cube.

Yeah, sorry, there was a diagram in the original question but all it showed was a cube split into 8 smaller cubes, 4 of which were dark and 4 of which were light.

How does it change my working that there is a silicon atom in the middle of every alternate cube, Face Diagonal should still be equal to 2 * Si-Si bond length, no? (Since there is an atom at the corner, and an atom in the centre, of each face).

[curiouscat]

What's the right answer?

 a / sqrt(3)

[Big-Daddy]

What's the right answer?

 a / sqrt(3)

Answer is a·(3^1/2/4) pm. No idea why it's not a·(2^1/2/2), they gave 1 mark out of 4 for this answer so clearly it is badly wrong.

[curiouscat]

What's the right answer?

 a / sqrt(3)

Answer is a·(3^1/2/4) pm. No idea why it's not a·(2^1/2/2), they gave 1 mark out of 4 for this answer so clearly it is badly wrong.

Yes that makes sense. Si-Si should indeed be
[tex]
\frac{\sqrt{3} a}{4}
[/tex]

I miscalculated before. Look along the body diagonal.

[curiouscat]

[Big-Daddy]

What's the right answer?

 a / sqrt(3)

Answer is a·(3^1/2/4) pm. No idea why it's not a·(2^1/2/2), they gave 1 mark out of 4 for this answer so clearly it is badly wrong.

Yes that makes sense. Si-Si should indeed be
[tex]
\frac{\sqrt{3} a}{4}
[/tex]

I miscalculated before. Look along the body diagonal.

Once we divide the unit cell into 8 cubes, the length of each cube in each direction is (1/2)a (which is why exactly 8 fit: a³/((1/2)a)³=8). The body diagonal of one of these smaller cubes is 3^1/2·(1/2)a. If there is an Si atom at the corner of this small cube and in the middle, then the distance between them, which is the Si-Si bond length, is half the body diagonal of the smaller cube, thus 3^1/2·(1/4)a.

But - where is the problem in my previous logic? Looks like two different Si-Si bond lengths are obtained depending on how you calculate it ...

[curiouscat]

By convention it is the shortest distance.

[Big-Daddy]

By convention it is the shortest distance.

Ah ok. I think they should have shown a diagram of the unit cell to make the connectivity clear, because then I wouldn't have wasted time calculating the bond length of a bond (corner to face-centre) that doesn't exist

[curiouscat]

Ah ok. I think they should have shown a diagram of the unit cell to make the connectivity clear, because then I wouldn't have wasted time calculating the bond length of a bond (corner to face-centre) that doesn't exist

I thought they explained it well enough. Problems don't come pre-packaged to make it easy for you to solve them.

You chose to ignore the relevant information.