Calculation of band gap energy from frequency vs wave-vector dispersion relation in 1D diatomic lattice

Question

In the experiment regarding modelling of 1D diatomic lattice via LC circuits,I was able to plot the dispersion relation of frequency vs wave-vector. As should be expected, I get a jump from the acoustic to the optic branch. We use harmonic approximation to model the lattice.

My question is how to calculate the energy band gap of this lattice from its Dispersion relation. $\theta$ is the phase(=wave-vector x lattice parameter 'a') and $\omega$ is 2$\pi$ x frequency. $$ \omega^2=K\Big(\frac{1}{M_1}+\frac{1}{M_2}\Big)\pm K\sqrt{\Big(\frac{1}{M_1}+\frac{1}{M_2}\Big)^2-\frac{4sin^2\theta}{M_1M_2}} $$

Answer 1

The band gap occurs at $\Theta=90^\circ$ , i.e. $\sin\Theta = 1$. This yields \begin{align} \omega_\pm^2 &= K \left( \frac{1}{M_1}+\frac{1}{M_2} \pm \sqrt{\left(\frac{1}{M_1}-\frac{1}{M_2}\right)^2} \right)\\ \end{align} If we w.l.o.g. assume that $M_1<M_2$ it follows \begin{align} \omega_+ &= \sqrt{\frac{2K}{M_1}}\\ \omega_- & = \sqrt{\frac{2K}{M_2}}\\ \Delta \omega &= \sqrt{2K}\left( \sqrt{\frac{1}{M_1}}- \sqrt{\frac{1}{M_2}}\right) \end{align}