Bragg condition and Bloch states

[Simobartz]

TL;DR Summary

I'm confused about why Bragg condition can be framed in the treatment of Bloch states, i need some extra context

I'm reading about Bloch states, these the are states of electrons in a periodic potential. What i know is that the electron in a Bloch state is shared between many ions and it is a stationary state.
However, for a 1-dimensional model I've read that at the edge of the first Brillouin zone, when when $k=\hbar \pi/a$, there is a finite discontinuity in the dispersion relation and this correspond to the Bragg condition which physically mean that an electron cannot propagate through the zone boundary and is reflected back.
I have a conceptual problem in understanding this idea, I've read on wikipedia about Bragg condition for a particle that hits a crystal while here there is nothing like that. For me an electron that is reflected back should hit something while here, of course, there is nothing on which the electron can bounce. Can you give me an idea about the concept/context behind these things? i would appreciate an almost math free answer because i think my problem is that I'm ignoring something important in the context.

 

[hutchphd]

Summary: I'm confused about why Bragg condition can be framed in the treatment of Bloch states, i need some extra context

cannot propagate through the zone boundary

Please describe what this (my annotation in red) means in more detail.

 

[Simobartz]

Thanks for replay, that is just a quote from my book. I'll write here the complete paragraph so that you have more context.

"The Bloch states provide us with a sound explanation for the existence of energy bands, which we conjectured using a qualitative argument about residence time, and in particular for the existence band gaps. Indeed, the energy of the Bloch states often take values only within some allowed intervals (the bands), which are often separated by forbidden regions (the band gaps). The behavior of $E$ versus $k$ shows then finite discontinuities at the "zone boundaries" where k is equal to a reciprocal lattice vector $\vec G$. But this is exactly the Bragg condition for the diffraction of all kinds of waves from a crystal, which physically means that an electron cannot propagate through the zone boundary and is in fact reflected back. Hence, $\vec k \rightarrow -\vec k$, and the momentum difference $2 \hbar \vec k$ is exchanged with the lattice."

I'd like to explain you better the situation but the point of the question is exactly that i don't understand what the author is speaking about. For me Bloch states are just stationary solution of the Schrodinger equation, i totally miss this "electron traveling and bouncing back" point of view.

 

[hutchphd]

Fair enough. I will write some words that may be on point or not.

1. The crux of electron propogation in a periodic potential is that electron momentum is not conserved because this background potential, tied to the hugely massive crystal lattice, can supply whatever momentum one wants.
2. Because of the periodicity, we can construct Bloch states to solve. The dispersion relation (E vs k) for these states looks much like a free particle but there are disallowed Energies (gaps) for certain k. We note that that these are related to the periodicity. In particular, in one dimension the gaps occur at $k=\frac {2\pi} a$ and multiples thereof. The physics is clear, at these "zone boundaries" the propagating electron is coherently reflected by each subsequent atom and any wave at this k must be a standing wave: no net propogation
3. In a 3D crystal we note that this is condition on the dispersion relation is identical to the condition for Bragg reflection of xrays. But this is to be expected, because the incoming light must resonantly match momentum and energy with the electron. An electron at the zone boundary is strongly coupled to the lattice and can therefore deliver the required rebound momentum to the entire crystal elastically for the appropriate xray.

One must always be wary of semiclassical explanations of quantum phenomena, but I think this explanation is fundamentally correct. The Bloch waves tell us that the only eigensolutions at the zone boundaries are standing waves (i.e. classically electrons bouncing back and forth)

 

[Simobartz]

Thanks again for replay. I think your words are on point because I've read something similar on my notes. Hoping to not bother you too much i will write for every point what i don't understand. I know for sure there is something that I'm missing out, i feel like I'm not looking at the problem from the correct point of view.

1. According to what i understand the electron in a Bloch state is stationary, so the momentum should be conserved as much as every measurable quantity. Should i consider this an approximation? Also, the periodic potential I've seen is static so it cannot transfer anything to the electron while time passes.
2. Form Bloch theorem it follows that in the first Brillouin zone there are alol the states, outside the first Brillouin zone the same states are repeated again.
   However we see a gap at the end of the first Brillouin zone, there is a "jump" of the energy at $k = 2 \pi / a$ . This imply that the energy of the states at $k <2 \pi / a$ is different (bigger) respect to the states at $k< 2 \pi / a$ ... how it is possible if the states at $k > 2 \pi / a$ are just a repetition of the states in the first Brillouin zone?
3. I don't understand what is meant by "zone boundary". From what I read it seems that $k = 2 \pi / a$ is the so called "zone boundary", but this is a boundary of a region of the reciprocal space. The electron lives in the direct space so it doesn't make much sense to spak about an electron reflected on a plane of the reciprocal space. Also I imagine the electron as diffuse state along all the N ions of the lattice and when you speak about the reflection of the electron and so on it seems we are speaking of different things.

 

[hutchphd]

These are good questions. Please look at the first several sections of wikipedia "Bloch's Theorem" . Seems pretty good to me with good pictures.

1. According to what i understand the electron in a Bloch state is stationary, so the momentum should be conserved as much as every measurable quantity. Should i consider this an approximation? Also, the periodic potential I've seen is static so it cannot transfer anything to the electron while time passes.
2. Form Bloch theorem it follows that in the first Brillouin zone there are alol the states, outside the first Brillouin zone the same states are repeated again.
   However we see a gap at the end of the first Brillouin zone, there is a "jump" of the energy at k=2π/a . This imply that the energy of the states at k<2π/a is different (bigger) respect to the states at k<2π/a ... how it is possible if the states at k>2π/a are just a repetition of the states in the first Brillouin zone?
3. I don't understand what is meant by "zone boundary". From what I read it seems that k=2π/a is the so called "zone boundary", but this is a boundary of a region of the reciprocal space. The electron lives in the direct space so it doesn't make much sense to spak about an electron reflected on a plane of the reciprocal space. Also I imagine the electron as diffuse state along all the N ions of the lattice and when you speak about the reflection of the electron and so on it seems we are speaking of different things.

(1) Momentum of the chunk of metal is conserved. Momentum of the electron is not, but the crystal momentum k (or quasimomentum of electron in this crystal) is "sort of" conserved. It is conserved to within a vector of the reciprocal lattice. This is dictated by the periodicity
2)For each value of k there is a ladder of energy states n which relate to the shape of the repeated atomic potential. We need only draw these in the first zone: the rest will just repeat because of (1). The wavelike envelope of the state is the same (same k) but the local detail changes according to n. The "jump" depends upon n but not k
3)This is bad nomenclature and confusing. As k reaches the zone boundary it becomes indistinguishable from -k (they differ by a reciprocal vector). This means in real space there is a resonant backscattering when the wavelength matches the a lattice spacing, so there is a standing wave. At other k the backscatter is not coherent. See Umklapp Scattering
All of these are nontrivial questions and important for underterstanding periodic structures. I recommend Ashcroft and Mermin Solid State Physics

 

[Simobartz]

Thanks for these answers, now I'm quite ok with point 2 and 3. I also read the wikipedia article as you suggested and it was indeed useful.

I still have a some doubt about point 1. For what i understand Bloch states are stationary states of the electron in a periodic potential. What do you mean with "crystal momentum" and "momentum of the chunk of metal", are they the same thing? I'm imagining something like the momentum of the ion plus electrons... i don't know.
Also, a Bloch state, $u_k(x) e^{i(kx- \omega t)}$, isn't a true plane wave and so it follows that its momentum is not exactly $p=\hbar k$ . It seems that you are referring to $p=\hbar k$ as the "crystal momentum", what do you mean?

 

[hutchphd]

Read again the section 3 of the Wikipedia article and try to think a little more about it. The solution is not a plane wave because the momentum of the electron is not a good quantum number but we define the "crystal momentum" which is and gives the Bloch wave: See section 3 and refine your question as needed.