Surface Energy of Miller Index Cross Sections

[intrepid_nerd]

ℽ = ½N_bερ_a
ℽ = surface energy
N_b = number of broken bonds
ε = bond strength
ρ_a = surface atomic density

ℽ_{100} = (½)( ² / _a² )(4)(ε)

ℽ_{110} = ( ⁵ / _sqrt(2) )( ^ε / _a² )

ℽ_{111} = (2sqrt(3))( ^ε / _a² )

the supplemental picture shows a cube with atoms at each corner and one atom in the middle of each facet of the full cube. the description for {100} says there are four broken chemical bonds, but what about the fifth atom in the center? for {110} is says there are five and for {111} it says three. WHAT?? if you picture the {110} it looks like 6 bonds are broken and for {111} it looks like five to me, am I hallucinating??
here is a link to make it easier to picture the sections: http://en.wikipedia.org/wiki/Miller_index"

PLEASE GIVE INSIGHT! THANKS A TON!

 

[BigJDubs]

The atoms in the center of each facet are not part of the Miller Index calculation. They are referred to as "dangling bonds" and they do not count towards the number of broken chemical bonds. For the {100} surface, there are four broken chemical bonds, for the {110} surface, there are five broken chemical bonds, and for the {111} surface, there are three broken chemical bonds.

 

[charng]

I can provide some clarification on the surface energy equations and the number of broken bonds for each Miller index.

Firstly, the equation for surface energy (ℽ) is a simplified version of the more general equation, which takes into account the angle of the surface and the bond angles between atoms. This simplified version assumes that all bonds are broken at a right angle and that all bonds have the same strength (ε).

Now, for the Miller indices, the number of broken bonds (Nb) is determined by the number of atoms on the surface that do not have a neighboring atom on the same surface. In the case of the {100} surface, there are four such atoms, as the fifth atom in the center is still surrounded by neighboring atoms on the same surface. This is why the equation for ℽ{100} only includes four broken bonds.

For the {110} surface, there are six atoms that do not have a neighboring atom on the same surface, thus resulting in a Nb value of six. Similarly, for the {111} surface, there are five such atoms, leading to a Nb value of five.

I understand that it may seem confusing when looking at the cube diagram, but it is important to remember that the Miller indices represent the orientation of the surface, not the actual number of atoms present. I hope this helps clarify any confusion.