Knowledge Bank

The frequency distribution of phonons in a solid is known to be exactly proportional in single phonon exchange processes to the double-differential neutron scattering cross-section $(d2\sigma /d\Omega dE)$ only for monatomic, cubic crystals which scatter incoherently.$\mathrm{<mrow\; data-mjx-texclass="ORD">1</mrow>}$ In the present work it is shown that for molecular crystals, an effective'' frequency distribution can be obtained directly from the double-differential scattering cross-section if the translational and rotational motions are assumed separable and specific potentials are used to describe the hindered rotations and translations. In this approach, the true frequency distribution is related to the effective'' distribution through a rotational effective mass which is momentum transfer dependent. This description is used to obtain the frequency distribution of ice. The differential scattering cross-section was measured for ice at $150\circ K$ using a beryllium-filter, time-of-flight spectrometer $(E0=40cm-1)$. Peaks are observed in the neutron spectrum at 50, 60, 72, 100, 145, 180, 204, 242, 280, 650 and $880cm-1$ in good agreement with Raman and infrared measurements. For a constant effective rotational mass of approximately 2 a.m.u., a frequency distribution is obtained from which the specific heat of ice is calculated. The results are in good agreement with the measured values of specific heat over the entire temperature range, $0-270\circ K$. Other thermodynamic quantities will be calculated to further test the correctness of the frequency distribution. A frequency distribution for water will also be presented. The authors are guests at the Army Materials Research Agency, Watertown, Mass. Permanent address of Sidney Yip: Nuclear Engineering Department, Massachusetts Institute of Technology, Cambridge, Mass.