- #1
Petar Mali
- 283
- 0
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}
\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}