= Long-wavelength inference of diatomic-chain parameters
For alternating <masses> $m_A,m_B$, nearest-neighbour spacing $a$ and equal <spring constants> $\alpha$, the zone-centre optical <angular frequency> and acoustic <group velocity> obey
$$
\omega_{\rm op}^2(0)=2\alpha\left(\frac1{m_A}+\frac1{m_B}\right),\qquad
v_s^2=\frac{2\alpha a^2}{m_A+m_B}.
$$
Therefore $\alpha=m_Am_B\omega_{\rm op}^2(0)/[2(m_A+m_B)]$ and $a=v_s\sqrt{(m_A+m_B)/(2\alpha)}$. These infer effective model parameters from a measured intercept and initial slope of a <phonon dispersion relation>. Fitting a real polar crystal to a nearest-neighbour scalar model does not guarantee that every measured point or its structural spacing is reproduced exactly.
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