Lennard-Jones potential and interatomic distance

[eneacasucci]

I've found this image online

(ref: https://edurev.in/t/188018/Origin-of-Energy-Bands ), it should be the graphical representation of the potential binding energy between two nearest-neighbor atoms.
I don't understand how it can be correlated to the Lennard-Jones pontential graph:

, in which we see the r_m (equilibrium distance) which is the most stable distance between the particles, that I assume being what in the previous figure is a.
1) Why in the first figure we see a minimum in the middle of the two atoms?
2) are the two potentials displayed related to different things?
3) does it mean that the distance to have the minimum potential is r_m and that V(r_m) is reached in the midpoint of the two particles?

 

[Charles Link]

I could be wrong, but I think the first curve represents the potential energy that the electron sees as a function of position as it moves around a couple of ions in a solid that is composed of positive nuclei plus an electron background.

Edit: It represents the periodic potential of an electron in a crystal in solid state physics. It can be used with the Schrödinger equation to solve for the wave function of an electron. Bloch's theorem is an important concept in this topic.

The Lennard-Jones potential is a graph of the potential energy of a diatomic molecule with off-setting electron charge around it as a function of the spacing of the nuclei.

Edit 2: If I remember my college studies of solid state physics correctly, in practice they use a pseudo-potential to represent approximately the periodic potential that the electron sees, (especially in the conduction band), as the nuclei are screened by the other electrons.

 

[Steve4Physics]

@eneacasucci, I'd like to add this to what @Charles Link has said.

... it should be the graphical representation of the potential binding energy between two nearest-neighbor atoms.

That's not what the graph shows. The graph shows the potential energy of an electron due to the electric field produced by a row of positive ions.

You might find it useful to sketch a graph of potential energy vs. separation for a single electron and single positive ion for yourself.

Consider this graph:

The graph shows the potential energy of an electron due to a row of positive ions (though only 3 ions are shown).

At P, the gradient is positive so the electron experience a force to the left; i.e. the electron is attracted to the ion labelled A.

At Q, the gradient is zero so the electron experiences no net force. That's because the electron exepriences equal mgnitude, opposite direction attractive forces which cancel.

At R, the gradient is negative so the electron experience a force to the right; i.e. it is attracted to the ion labelled B.

The Lennard-Jones graph is different . It shows the potential energy of a pair of neutral atoms (or molecules) as a function of separation . Note that atoms repel at close distances and attract at long distances; this is very different from an electron and positive ion which always attract.

 

[eneacasucci]

@eneacasucci, I'd like to add this to what @Charles Link has said.


That's not what the graph shows. The graph shows the potential energy of an electron due to the electric field produced by a row of positive ions.

You might find it useful to sketch a graph of potential energy vs. separation for a single electron and single positive ion for yourself.

Consider this graph:
View attachment 364656
The graph shows the potential energy of an electron due to a row of positive ions (though only 3 ions are shown).

At P, the gradient is positive so the electron experience a force to the left; i.e. the electron is attracted to the ion labelled A.

At Q, the gradient is zero so the electron experiences no net force. That's because the electron exepriences equal mgnitude, opposite direction attractive forces which cancel.

At R, the gradient is negative so the electron experience a force to the right; i.e. it is attracted to the ion labelled B.

The Lennard-Jones graph is different . It shows the potential energy of a pair of neutral atoms (or molecules) as a function of separation . Note that atoms repel at close distances and attract at long distances; this is very different from an electron and positive ion which always attract.

This explanation is super informative and accurate, I cannot thank you enough for this. My notes were wrong and I was so confused about it.