Calculating Integral of Cubic Lattice Sum

[Petar Mali]

If I have cubic lattice and need to calculate sum

$$\frac{1}{N}\sum_{\vec{k}}\frac{1}{\sqrt{1-\gamma^2(\vec{k})}}coth\frac{6SI\sqrt{1-\gamma^2(\vec{k})}}{2T}$$

where

$$\gamma(\vec{k})=\frac{1}{3}(cosk_xa+cosk_ya+cosk_za)$$

I must go from sum to integral

$$\frac{1}{N}\sum_{\vec{k}}F(\vec{k})=\frac{a^3}{(2\pi)^3}\int F(\vec{k})d^3\vec{k}$$

My question is what is lower limit and upper limit in this integral. Is it perhaps
$$k_x$$ goes from $$-\frac{2\pi}{a}$$ to $$\frac{2\pi}{a}$$


$$k_y$$ goes from $$-\frac{2\pi}{a}$$ to $$\frac{2\pi}{a}$$

$$k_z$$ goes from $$-\frac{2\pi}{a}$$ to $$\frac{2\pi}{a}$$

 

[mathman]

It depends on the k(vector) range in the original sum formula.

 

[Petar Mali]

This is box quantisation and integral is over first Brillouin zone. Am I right about range of integration?

 

[kanato]

Your limits for a cubic lattice should go from 0 to 2pi/a or -pi/a to pi/a.