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

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. $$\omega^2=K\Big(\frac{1}{M_1}+\frac{1}{M_2}\Big)\pm K\sqrt{\Big(\frac{1}{M_1}+\frac{1}{M_2}\Big)^2-\frac{4sin^2\theta}{M_1M_2}}$$ • M1 and M2 are the masses of the two different atoms. Feb 3 '17 at 8:52
• $\omega = 2\pi\nu$, and $E=h\nu$, aren't they? Feb 3 '17 at 16:38
• Since harmonic approximation is used, and energy of a harmonic oscillator is quantised=$E_n=\hbar\omega(n+\frac{1}{2})$, I was not sure whether to use this formula. Feb 5 '17 at 10:42

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}
• Depends on what you mean by energy. The energy of one phonon with frequency $\omega$ is $E=\hbar \omega$. Feb 3 '17 at 16:50