# Group velocity in different dimensions

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?

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