Energy Levels in a Crystalline Solid — Question 2

Question 2

A periodic chain turns discrete wave numbers into a narrow band

A one-dimensional ring has N equally spaced sites, spacing a, and length $L=Na$. A supplied dispersion is $E(k)=E_0-2t\cos(ka)$ with $t>0$. Periodicity requires $e^{ikL}=1$.

1. Find the allowed k spacing and band width.

2. Count the states and electrons needed to fill this single band, including spin.

Question 2 - Solution

A periodic chain turns discrete wave numbers into a narrow band

(a) Method and calculation

$$k_jL=2\pi j\Rightarrow\boxed{\Delta k=2\pi/(Na)}.$$$$E_{\min}=E_0-2t,\ E_{\max}=E_0+2t\Rightarrow\boxed{W=4t}.$$

(b) Interpretation and further calculation

$$\text{One reciprocal interval of width }2\pi/a:\quad\frac{2\pi/a}{\Delta k}=N.$$$$\boxed{N\text{ spatial states},\quad2N\text{ electron slots}}.$$

Physical check

Wave numbers separated by a reciprocal lattice vector describe the same Bloch label. A finite chain samples the band; exact attainment of both band extrema depends on the allowed k values.