# GaAs Phonon Dispersion Spectrum understanding

I have a beginner question on solid state physics:

GaAs has phonon dispersion spectrum that looks so:

the horizontal axis parametrizes the wave numer values $$k$$, therefore it seems plasible that the direction axis of $$\vec{k}$$ is fixed in the whole diagram. In other words $$k=\vert \vec{k} \vert$$.

On the other hand $$\Gamma, X, U, K, L$$ represent different directions in the $$k$$-space: see here

Therefore I'm a bit confused what is parametrized by horizontal axis.

Or does every connected path in the diagram correspond to a special direction $$\frac{\vec{k}} {\vert \vec{k} \vert}$$ for example $$X, K$$ or $$L$$? If it is so, then which path belongs to which direction $$\frac{\vec{k}} {\vert \vec{k} \vert}$$?

Could somebody explain how to read/understand what is going on at the parametrization of the $$k$$-axis? What do the big letters on the bottom say? $$L - \Gamma - X - U,K - \Gamma$$.

Does it mean that firstly we start at point $$L$$ of the $$\vec{k}$$-space and go along the straight line to $$\Gamma =(0,0,0)$$ (thus along the direction $$\vec{k}_{\Gamma}- \vec{k}_L$$), then - after arriving at $$\Gamma$$, we change the direction to $$\vec{k}_X -\vec{k}_{\Gamma}$$ an go along it until we arrive point $$X$$ and so on?

What happens between $$X$$ and $$U,K$$?

• Between, say, the [100] and [111] directions, you have wave vectors of [1xx] with x increasing from 0 to 1 along the segment joining those two particular directions. Commented Oct 27, 2019 at 3:30
• ok, so on every partial intervall between two big letter say A and B on the $k$-axis we choose as direction $\frac{\vec{k}} {\vert \vec{k} \vert}$ the difference vector of direction vectors of $A$ and $B$,right? Commented Oct 27, 2019 at 22:46
• The graph shows dispersion along high-symmetry directions in the Brillouin zone. There are many other points in k-space, with other directions. Densities of states are derived from sampling all of the Brillouin zone (sampling an irreducible wedge is enough).
– user137289
Commented Oct 27, 2019 at 23:17
• yes yes, that's clear. The point is for example if well consider the partial intervall between $X$ and $U$ in the middle, in which direction in the $k$-space we are going there? As you said $X$ and $U$ represent certain high-symmetry directions in $k$-cpace, let call them $\vec{k}_X$ and $\vec{k}_U$. My intuitive suspicion is that as long as we are going along path X - U on the $k$-axis, we go in direction $\vec{k}_U- \vec{k}_X$. Is this the correct interpretation or did I misunderstood the point? Commented Oct 27, 2019 at 23:29
• The cuts are along a specific direction. E.g. $\Gamma$ - $X$ is along the direction from $\Gamma$ towards $X$. etc. Commented Nov 10, 2019 at 5:49

The purpose of drawing the dispersion relation like this is to facilitate the visualization of a 3-dimensional function in a 2-dimensional plot by traveling in lines along high symmetry points. Traveling along 2 points, say from $$\Gamma$$ to $$X$$ means to travel from k value [0,0,0] to k value [1,0,0] with points of the form [x,0,0] in between.