Re: [Phys-l] Band splitting in Carbon (diamond)...

[Jeff Loats <jeff.loats@gmail.com>]

Hi Stefan,

Thanks for your very helpful comments here!

On Mon, Mar 1, 2010 at 3:19 PM, Stefan Jeglinski <jeglin@4pi.com> wrote:

> As you bring many many atoms together to form a bulk crystal, you can
> think of this happening on a finer and finer level. The energy levels
> in a band are still distinct, but very finely separated. In solid
> state physics, the assertion of periodicity in the Hamiltonian,
> V(x+a) = V(x), results in a new quantum number, usually denoted k,
> the wavevector. The band diagrams you see are energy plots as a
> function of k, E(k).

I am totally with you here... glad to have that part of my thinking
verified, at least.


> > I am having trouble finding a good sophomore-level description of what
> > causes the splitting to occur. That is, why don't the 2s and 2p bands just
> > continue to overlap, forming one huge band?
>
> They in fact do. And the band *is* huge (but be careful was how you
> envision "huge" - you are generally working in k-space, not
> x/y/z-space). There is overlap, but no 2 electrons can have the same
> quantum numbers. Not a problem, as you have a new quantum number
> (wavevector) for every 2 (spin up/down) electrons. The valence band
> is typically the set of highest energies occupied, "derived" from the
> HOMO, "derived" from the outer occupied shell of the atom in
> question. The conduction band is the set of lowest unoccupied
> energies, "derived" from the LUMO, "derived" from the outer
> unoccupied shell of the atom in question.
>
> >  What is the mechanism for the later split.
>
> Interaction. But the "splitting" occurs as soon as the interaction
> does. The split is not later. There is always an energy "gap" between
> any energy levels, by definition. The ultimate presence of "bands" is
> a convenient way to picture things when a large number of atoms have
> been brought together in periodic fashion.
>
> My explanation is somewhat sophomoric as well, and there will be
> plenty of quibbles, but the essence is there I think. Interactions
> mix wavefunctions, and when they are mixed, they "move" to different
> energy levels (different soln to Schrodingers eqn). As you bring
> atoms together, the interactions increase, further changing the
> energy levels until they can be visualized as bands. The notion of
> periodicity (see Bloch's theorem) led to a brilliant way (k-space) to
> view bulk solids in the form of crystals. Amorphous solids
> (non-crystalline) makes for messier analysis. On the level you are
> trying to teach it, stick with crystals. But you will have to invoke
> quantum mechanics, I think.

Invoking QM is fine... I meant sophomore in terms of physics majors in their
2nd year, not pejoratively.

In my response to JD I refined what I am asking a bit, trying to be clear
that it isn't the band formation, nor the overlap into a "superband" that I
am fuzzy on, it is the later (as in, as r continues to shrink) appearance of
a significant gap between two "bands" which are now not recognizable as 3s
or 3p (at least, not in terms of the number of states).

Does that help clarify my question?

Thanks again for your response on this, it helped me solidify several
things. Your comments are much appreciated.

Cheers,

Jeff