I am given the following equation for the group velocity in one-dimension:

$$v_g = \frac{d\omega}{dk}.$$

In solid state physics one has $-\pi/a \leq k \leq \pi/a$ for a one-dimensional lattice with atoms where $a$ is the equilibrium distance between atoms. But my questions are: Is $k$ in the equation of the group velocity the absolute value of $k$ so rather $v_g = \frac{d\omega}{d|k|}$ and how can we generalize this to 3D for example?

  • $\begingroup$ In 3D, $v_g$ will be a vector quantity, just like $k$ $\endgroup$ – By Symmetry Jan 14 at 13:34
  • $\begingroup$ And what is $k$ in the 1D case? Is this the absolute value? $\endgroup$ – Dani Jan 14 at 13:37
  • $\begingroup$ no. In the 1D case $k$ is just $k$ $\endgroup$ – By Symmetry Jan 14 at 13:43

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