6.15: The first Brillouin zone

Since there are only N distinct values of the coefficients (corresponding to one period of the Fourier transform), we typically restrict k to the N values in the range \(-N/2 < n \leq N/2\), i.e.

\[ -\frac{\pi}{a_{0}} < k \leq \frac{\pi}{a_{0}} . \nonumber \]

This is known as the first Brillouin zone. Other values of k are either not permitted by periodic boundary conditions, or \(c(x)=\text{exp}[ikx]\) reduces to one of the N solutions. For example, consider \(k = 2\pi (n+N)/L\):

\[ c(x) = e^{i\frac{2\pi (n+N)}{L}x} = e^{i\frac{2\pi (n+N)}{Na_{0}}ra_{0}} = e^{i\frac{2\pi n}{N}r+i2\pi r} e^{i\frac{2\pi n}{N}r} = e^{i\frac{2\pi n}{L}x} \nonumber \]

where \(a_{0}\) is the spacing between unit cells.

Figure \(\PageIndex{1}\): A molecular orbital is described by linear combinations of the wavefunction of the unit cell. The coefficients, \(c_{r}\), are phase factors. The phase coefficients are discrete – there are only N of them. Thus, the Fourier transform of the coefficients contains only N unique values (it is periodic). We can restrict the range of k values without losing information. Typically, we chose k values in the first Brillouin zone (\((-\pi /a_{0} < k \leq \pi /a_{0})\)). Note also that the application of periodic boundary conditions fixes the spacing between k values at \(2\pi /L\).