Coherent neutron scattering phonons

Coherent neutron scattering is normally used to investigate collective excitations such as phonons in crystals. It has not been applied extensively to liquid crystals, but... 

Figure 4 Schematic vector diagrams illustrating the use of coherent inelastic neutron scattering to determine phonon dispersion relationships, (a) Scattering m real space (h) a scattering triangle illustrating the momentum transfer, Q, of the neutrons in relation to the reciprocal lattice vector of the sample t and the phonon wave vector, q. Heavy dots represent Bragg reflections.

If the displacements of the atoms are given in terms of the harmonic normal modes of vibration for the crystal, the coherent one-phonon inelastic neutron scattering cross section can be analytically expressed in terms of the eigenvectors and eigenvalues of the hannonic analysis, as described in Ref. 1. 

Coherent Inelastic Scattering.—Inelastic neutron collisions with the solid can excite phonon modes (collective vibrations) and if the coherently scattered component can be detected variation with direction within the solid, i.e. the phonon dispersion curve, can be determined. This technique is well established for bulk solids and has been used recently to examine the properties of small particles (carbon black). 

Fig. 5.8 shows the experimental phonon dispersion relations of a perdeuterated naphthalene crystal (N-dg) determined at a temperature T = 6 K by inelastic neutron scattering [7]. Deuterium (D) has a much higher scattering cross section for coherent inelastic neutron scattering than the isotope H. Nevertheless, to gain a... 

Table 4.1-9 Phonon frequencies at symmetry points. Diamond (C) (v in THz, 300 K, from Raman spectroscopy) silicon (Si) (v in THz, 296 K, from inelastic neutron scattering) germanium (Ge) (v in THz, 300 K, from coherent inelastic nentron scattering) gray tin (a-Sn) (v in THz, 90 K, from inelastic thermal neutron scattering) silicon carbide (3C-SiC) (phonon wavenumbers v in cm , RT, from Raman spectroscopy) silicon carbide (6H-SiC) (v in THz, derived from photoluminescence data)...

Table 4.1-59 Phonon wavenumbers of gallium compounds. Gallium nitride (GaN), T = 300K, from Raman spectroscopy gallium phosphide (GaP), RT, from an analysis of Raman, neutron, luminescence, and absorption data gallium arsenide (GaAs), T = 296 K, from coherent inelastic neutron scattering gallium antimonide (GaSb), T = 300 K, from second-order Raman effect...

Table 4.1-80 Phonon wavenumbers of indium compouuds. Indium nitride (InN), T = 300 K indium phosphide (InP), RT, from coherent inelastic neutron scattering, carrier concentration n = 10 em indium arsenide (InAs), from Raman scattering indium antimonide (InSb), T = 300 K, from inelastic neutron scattering, carrier concentration n = SxlO cm ...

Inelastic scattering of neutrons is caused by an oscillatory motion. An example is the inelastic scattering of neutrons by phonons. Following a textbook [10], the coherent double-differential scattering cross-section of neutrons for one phonon process is given by... 

Patterson and Lynn [6] have reported a lattice dynamical study of the host lattice CsjSiFg based on neutron scattering, Raman, and infrared absorption measurements. Dispersion relations for phonons with energies less than 160 cm have been determined along three symmetry directions by coherent inelastic neutron scattering experiments. In Fig. 7 the photon dispersion results for Cs2SiFg are shown [6] in which the experimental data are represented by circles. The solid lines correspond to dispersion curves calculated with a rigid-ion, lattice dynamical model. 

Here is the momentum transferred from the neutron and x is one of the reciprocal lattice vectors of the palladium lattice. Thus, the incoherent scattering sees all the vibration modes but the coherent scattering selects one particular phonon for a particular experimental value of Q. It is now clear that both the incoherent and coherent one-phonon scattering will depend on the shape of the optical dispersion curves and hence will be influenced by hydrogen-hydrogen interactions. Indeed, one of the first observations of inelastic scattering from a hydride [10] interpreted the shape of the optical peak in terms of a frequency distribution broadened by H-H interactions. 

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