What Is the Shortest Reciprocal Vector for a BCC Lattice?

[malawi_glenn]

Homework Statement

Find the shortest reciprocal vector G, given below, v_1,...,v_3 are integers.

$$\vec{G} = \frac{2 \pi}{a}\left( (v_2 + v_3 )\vec{x} + (v_1 + v_3 )\vec{y} + (v_1 + v_2 )\vec{z} \right)$$

Homework Equations

x,y,z are ortonogal, length 1

$$l = l(v_1, v_2, v_3) = \vert \vec{G} \vert = \sqrt{\vec{G}\cdot \vec{G}}$$

$$l = \sqrt{ (v_2 + v_3 )^{2} + (v_1 + v_3 )^2 +(v_1 + v_2 )^2 }$$

The Attempt at a Solution

I want to minimize l(v_1, v_2, v_3)

$$\dfrac{\partial l}{\partial v_1} = \dfrac{2 \pi \left( (v_1 + v_3 ) + (v_1 + v_2 ) \right) }{\sqrt{ (v_2 + v_3 )^{2} + (v_1 + v_3 )^2 +(v_1 + v_2 )^2 }} = 0$$

etc. Gives me following linear equation system, it has only trivial solutions

$$\left( \begin{array}{ccc|c} 2 & 1 & 1 & 0 \\ 1 & 2 & 1 & 0 \\ 1 & 1 &2 & 0 \end{array}\right)$$

v_1 = v_2 = v_3 = 0

And that is not true, they should be something like

$$\frac{2 \pi}{a} \left( \pm \vec{x} \pm \vec{y} \right)$$

etc.


Now what have I do wrong

by the way, this is the general reciprocal lattice vetctor for bcc lattice. I want to construct the first Brillouion zone.