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Consider a monoatomic 1-D chain of atoms (only acoustic branch). What happens with the inference of the dispersion curve through neutron scattering? In one dimension, conservation of momentum dictates $$ k'=k+K_s $$, if k_s is the phonon momentum vector and we only consider processes where a phonon is absorbed by the neutron. The corresponding omega you get from conservation of energy is thus $$ \omega = \frac{\hbar}{2 m_n} (2 k K_s+K^2_s) $$, at odds with the usual dispersion relation for the acoustic branch. What went wrong?