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I have a beginner question on solid state physics:

GaAs has phonon dispersion spectrum that looks so:

enter image description here

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 $U,K$?

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  • $\begingroup$ 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. $\endgroup$
    – Jon Custer
    Commented Oct 27, 2019 at 3:30
  • $\begingroup$ 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? $\endgroup$
    – user267839
    Commented Oct 27, 2019 at 22:46
  • $\begingroup$ 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). $\endgroup$
    – user137289
    Commented Oct 27, 2019 at 23:17
  • $\begingroup$ 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? $\endgroup$
    – user267839
    Commented Oct 27, 2019 at 23:29
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    $\begingroup$ The cuts are along a specific direction. E.g. $\Gamma$ - $X$ is along the direction from $\Gamma$ towards $X$. etc. $\endgroup$
    – Nathan Holman
    Commented Nov 10, 2019 at 5:49

1 Answer 1

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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.

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