# Radius for a unit cell

1. May 19, 2010

### cotyledon

What is the average radius for a unit cell, for example, for Na? Is it the average of the ionic radii for the different coordination numbers (weighting each coordination number the same)? Or would it be the covalent radii or something else?

2. May 19, 2010

### alxm

IIRC, metallic sodium is a body-centered cubic (BCC) crystal. The unit cell looks like http://en.wikipedia.org/wiki/File:Lattice_body_centered_cubic.svg" [Broken], a cube with side a, with one atom in the middle and the neighboring atoms on the 8 corners.

So the diagonal is 4 radiii in length, so if the side of the unit cube is a you have $$4r = \sqrt{a^2+a^2+a^2} = \sqrt{3}a$$ and the radius is $$r = \frac{\sqrt{3}a}{4}$$.
There are two whole atoms in a unit cell (one in the middle and 1/8 in each corner), so the unit cell weighs $$\frac{2*M_{Na}}{N_A}$$ g. The density of sodium metal is 968 kg/m3.

Do the arithmetic and you get a radius of 1.86 Å. That's called the 'metallic radius'.

As you can see, the size of the unit cell depends on the type of unit cell, the type of bonding going on, etc. Pick up a book on crystallography and you'll see it can actually get rather complicated.

Last edited by a moderator: May 4, 2017