Crystal structures and number of bonds of each atom

[Helena Wells]

Is there a general rule for a crystal structure how many bonds each atom in the crystal lattice will make ?For example Si has a face centered cubic structure so it makes 4 bonds regardless of the fact Si has 4 valence electrons?

 

[etotheipi]

You just count the number of nearest neighbours, no? Silicon has a face-centred cubic structure with a two atom motif $\{ (0,0,0), (\frac{1}{4},\frac{1}{4},\frac{1}{4}) \}$ attached to each lattice point, so the coordination polyhedra in that structure are tetrahedral.

 

[Helena Wells]

You just count the number of nearest neighbours, no? Silicon has a face-centred cubic structure with a two atom motif $\{ (0,0,0), (\frac{1}{4},\frac{1}{4},\frac{1}{4}) \}$ attached to each lattice point, so the coordination polyhedra in that structure are tetrahedral.

What do you mean by a 'two atom motif'?I don't know crystal structures very well.

 

[Helena Wells]

You just count the number of nearest neighbours, no? Silicon has a face-centred cubic structure with a two atom motif $\{ (0,0,0), (\frac{1}{4},\frac{1}{4},\frac{1}{4}) \}$ attached to each lattice point, so the coordination polyhedra in that structure are tetrahedral.

Do all chemical elements which its solid crystal structure is face-centered cubic its atoms make 4 covalent bonds?

 

[etotheipi]

What do you mean by a 'two atom motif'?I don't know crystal structures very well.

Ah, yes sorry, I should have explained better.

A lattice is a set of purely mathematical points in space such that every lattice point is exactly equivalent (that last part is an important constraint - you only get 14 such "Bravais" lattices). They're generated by translations $\mathbf{R}_{UVW} = U \mathbf{a} + V \mathbf{b} + W \mathbf{c}$ for integers $U, V, W$. The vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are called primitive vectors, or basis vectors.

You get a crystal structure by convoluting the lattice with a certain "motif". A motif is just a set of atoms which you attach to every lattice point. A motif could be as simple as a single atom, or as complicated as hundreds of atoms attached to every lattice point (complicated biological structures, for example!).

So it's important to distinguish between the lattice (purely mathematical set of points), and the crystal structure (lattice convoluted with actual stuff).

Do all chemical elements which its solid crystal structure is face-centered cubic its atoms make 4 covalent bonds?

I don't think so, no. For example $\gamma$-iron has a coordination number of 12, I think, but has an underlying FCC lattice.

Also, I'm not sure if it's best to say that the atoms are covalently bonded to each other, at least not in the general case. Perhaps someone else can advise

 

[Helena Wells]

Ah, yes sorry, I should have explained better.

A lattice is a set of purely mathematical points in space such that every lattice point is exactly equivalent (that last part is an important constraint - you only get 14 such "Bravais" lattices). They're generated by translations $\mathbf{R}_{UVW} = U \mathbf{a} + V \mathbf{b} + W \mathbf{c}$ for integers $U, V, W$. The vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are called primitive vectors, or basis vectors.

You get a crystal structure by convoluting the lattice with a certain "motif". A motif is just a set of atoms which you attach to every lattice point. A motif could be as simple as a single atom, or as complicated as hundreds of atoms attached to every lattice point (complicated biological structures, for example!).

So it's important to distinguish between the lattice (purely mathematical set of points), and the crystal structure (lattice convoluted with actual stuff).
I don't think so, no. For example $\gamma$-iron has a coordination number of 12, I think, but has an underlying FCC lattice.

Also, I'm not sure if it's best to say that the atoms are covalently bonded to each other. Perhaps someone else can advise

Ok thanks!In crystal structures made of 1 chemical element they are covalently bonded.

 

[etotheipi]

Yeah sorry I couldn't be of more help haha, maybe someone else can add something. There's a few chapters in Ashcroft/Mermin about crystal structures that might be worth taking a look at if you're interested.

 

[Baluncore]

Is there a general rule for a crystal structure how many bonds each atom in the crystal lattice will make ?For example Si has a face centered cubic structure so it makes 4 bonds regardless of the fact Si has 4 valence electrons?

If that was the case NaCl would fall apart due to lack of bonds.
It will depend on the bond type.

An ionic bond will share the available electrons with the cloud and so will pack based on the ionic radii of the elements present. Pure metals tend to crystalise in a close packed maximum density arrangement.

A covalent bond within a molecule will pass electrons along the molecular network, tunneling across gaps etc. But those covalent molecules will crystalise in structures dictated by the topological packing of adjacent molecules, held together by weaker forces.

 

[DrDu]

The crystal structure of silicium is also called the diamond structure, as like in diamond, each atom has four nearest neighbours with which it shares 4 covalent bonds.
Furthermore, while you may call bonds in substances made up from only one element covalent, in the case of metals, these bonds will be highly electron deficient and not of the usual 2 electron per bond type. They are therefore called "metallic bonds" or "delocalized bonds" and the number of next neighbours is usually much higher in metals than the number of valence electrons.