# Atoms as spheres in packing fraction of crystal lattice

1. May 23, 2015

### spaghetti3451

Why are atoms taken to be spheres, and not of some other shape, in the calculation of the packing fraction of different crystal lattices?

In other words, what experimental evidence and theoretical reasoning motivates this form of the atomic shape?

2. May 24, 2015

### ash64449

Firstly.. i consider we cannot give atoms any shape due to uncertainty principle.

Earlier, Atoms were considered to be spherical because orbits of last electron was considered to be spherical. but we have to rule that out since there is no definite path for electron. It has been replaced by the statements of probability which is dictated by the wave functions of the last electron.

3. May 24, 2015

### spaghetti3451

This is the sum total of the views I've gathered from the internet:

Atom is described spherical in shape just for the sake of simplicity. It's very similar to (not the same as) a group of honey bees (electrons) swarming over a flower (nucleus). The truth is that you can only assign a probability density function for the positions of the electrons around the nucleus, which is in turn based on the solution of the Schrodinger equation for the Coulombic potential under consideration. These give rise to various so-called orbitals which delimit the region in which $95\%$ of an infinite set of measurements of the positions of the electrons is likely to provide. The "shape" of the atom is determined by this outer boundary.

Now, having precisely defined the meaning of "the shape of an atom", we can rigorously state when an atom is indeed "spherical":

Most atoms, when they are free (and of course most elements are not found as free atoms in nature), are not spherical. The conditions for an atom to be spherical are any of the following:

• The atom has electrons only in the s orbitals.
• The atom has the sub-shell of its largest principal quantum number either half-filled or full-filled (Unsold's Theorem).

That does not, in any way, make it clear why atoms are considered to be spheres in the calculation of packing fractions. I guess it's only because the spherical shape is the simplest and most intuitive model of an atom.

Last edited: May 24, 2015
4. May 24, 2015

### ash64449

yes you are right.... makes sense

5. May 25, 2015

### keV.92

Just wanted to put in my two cents on this one, given that I've looked into the topic before.

What it really comes down to is the need of Materials Engineers and Scientists to compare values such as mass-density or line-density at the atomic scale. The lengths are generally large enough that we don't need to worry about detailed modeling of the orbitals, so we approximate the outermost electrons as spheres. This allows the massive amount of data on crystal structures to be comparable because it's all using the same, generally accurate method.

I'd also point out that methods like X-Ray Diffraction are actually looking at the internuclear distance, and do not involve any interaction with the electron lattice. These positions are essentially fixed, down to the picometer; materials scientists generally don't care about the resultingly small uncertainty in the location of the lattice points.

This is an old model of course, but it works quite well for modelling the mechanical and thermodynamic properties of materials. If you're interested, one of my favorite youtube videos shows Dr. Bragg himself explaining dislocations by way of the sphere model. With bubbles.