Chemistry of solids: allowed wavevectors
In a crystal, electron waves are reflected off the periodic lattice and this (Bragg) reflection is the cause of the energy gaps also called a band structure. If there is a weak periodic potential in a 1D crystal then the energy gaps occur at wavevector $k=\pm \pi/d$ where d is the lattice constant. 

If $\bar G$ represents the reciprocal lattice vector then the general Bragg condition for diffraction of a wave with wavevector $\vec K$ can be written as 
$(\vec k + \vec G)^2=k^2 $ 

In one dimension this condition becomes 

$$k=\pm \frac{1}{2}G = \pm n\pi/d$$ 

and $G=2\pi n/d$ is a reciprocal lattice vector and $n=1,2,3..$  an integer. This is where your factor of 2 appears.

The reflection at $k=\pm \pi/d$ occurs because one wave is reflected of an atom and interferes constructively with phase difference $2\pi$ with one reflected from the nearest neighbour atom. The Brillouin zone is the $\bar k$ space between $-\pi/d$ and $+\pi/d$.

(The reciprocal lattice vectors have the form $\vec G =h\vec A+k\vec B + l\vec C$ for integers h, k, l. The vectors are defined as 

$$\vec A= 2\pi \frac{b~ \times~ c}{a\cdot b~ \times~ c}~~
\vec B= 2\pi \frac{c~ \times~ a}{a\cdot b~ \times~ c}~~
\vec C= 2\pi \frac{a~ \times~ b}{a\cdot b~ \times~ c}$$
for primitive lattice vectors a, b, c)