Cubic Lattice Atom Diameter Calculation

[Bob Ho]

Homework Statement

Barium metal crystallizes in a body-centred cubic lattice. The density of the metal is 3.50gcm^-3. Calculate the radius(in pm) of a barium atom.

M(Ba)=137.3g/mol , NA = 6.022x10^23/mol

The Attempt at a Solution

For 1 unit cell: m=2x137.3/6.022x10^23
=4.56x10^-22g

V=m/p
=4.56x10^-22/3.5
=1.303x10^8pm^3

Cell side length a=v^(1/3)=506.9pm

From here I am confused, the answer is that the cell diagonal length =4r=√3a
Atomic radius r=√3/4a=219.5pm

Can anyone please explain to me how the cell diagonal length varies with different structures, or anything that can help me, thanks.

 

[Borek]

It is a (relatively) simple geometry. General approach is to find a plane that goes through atoms centers, draw the atoms and see what is known. You will usually end with some mix of triangles and rectagles or parallelograms where you know some of the distances and you have to find others.