# Fermi surface questions

In order to define the fermi surface we must need to know about the momentum space. But I found a little bit about momentum space. Can you elaborate it pleaese? what is the meaning of the line

the k-vector has dimensions of reciprocal length, so k is the frequency analogue of r, just as angular frequency ω is the inverse quantity and frequency analogue of time t.

Suppose you have a signal that varies sinusoidally in time. We can write $f(t) = \sin(\omega\, t)$, where $\omega$ is the angular frequency of the signal. Similarly we can have a wave that is moving through space, $f(x) = \sin( k\,x)$. Here the value $k$ tells us how quickly the signal moves from one maximum to the next when $x$ changes -- the so-called spatial frequency. For a more general travelling wave in 3-dimensions (that has a sinusoidal nature), you can write
$f(\mathbf{r},t) = \sin\, (\mathbf{k}\cdot\mathbf{r} - \omega\, t)$.
Instead of travelling in the $x$ direction, this wave is travelling in the $\hat{\mathbf{k}}$ direction, has spatial frequency $|k|$ and temporal frequency $\omega$. You can find the velocity of the wave-fronts by looking at points that satisfy $|k|x - \omega\, t = \mathrm{const}$.