A few simple questions about band structure and conduction

1. Mar 29, 2013

VortexLattice

Hi! I'm reading about band structure and conduction in regular crystals and semiconductors, and I've hit a few confusing points my book doesn't explain well enough for me.

In one part, they're explaining conduction in a 1D crystal with lattice spacing $a$ (and therefore reciprocal lattice vectors $G_n = 2n\pi/a$), and are showing the origin of band gaps. They give this diagram:

as the (extended/reduced/repeated) band structure. Then they say

(where the Bragg condition is $n\lambda = 2a sin(\theta)$ just like in a normal 3D crystal, but with $\theta = \pi/2$ for the 1D case.)

Here's where I'm confused: First of all, I don't know why "scattering admixes components with wave number $k + G_n$ for all $n$." But I'll trust them on that, it seems reasonable. Secondly, they say that "in general the added components have different energy, so mixing is weak. It becomes strong only when the waves are of equal energy, which determines $k$."

So then they set $E(k) = E(k + G_n)$ and solve for $k$ to find that it happens at $k = G_n/2$, which agrees with the Bragg condition. But, I'm confused. I thought, because of the nature of the Brillouin zone, $k + G_n$ has the same properties as $k$ for any $k$. In fact, if you look at the repeated band structure, it really looks like if you choose any $k$ and hop over $G_n$, you're at the same height and thus have the same exact energy. So I'm a little confused why this condition isn't satisfied for any $k$.

My second question regards Bloch Oscillations. The book shows that electrons in an applied E field will oscillate rather than be accelerated uniformly. However, they say that electrons in a partly filled band will conduct, because they will be scattered to other states long before an oscillation is complete. So my question is: Would the electrons in a partly filled band not conduct if the crystal had no impurities and was very cold?

My last question is regarding this paragraph:

I'm not sure why "an odd number of electrons per atom results in a half-filled top band", or really what they mean by "top band". I actually get the concept they're talking about, with the Fermi level, but not that sentence (/how it relates to the above band diagram I posted).

Any help would be much appreciated! Thanks!