Inelastic neutron scattering due to acoustic vibrations confined in nanoparticles: Theory and experiment

The inelastic scattering of neutrons by nanoparticles due to acoustic vibrational modes (energy below 10 meV) confined in nanoparticles is calculated using the Zemach-Glauber formalism. Such vibrational modes are commonly observed by light-scattering techniques (Brillouin or low-frequency Raman scattering). We also report high-resolution inelastic neutron-scattering measurements for anatase ${\text{TiO}}_2$ nanoparticles in a loose powder. Factors enabling the observation of such vibrations are discussed. These include a narrow nanoparticle size distribution which minimizes inhomogeneous broadening of the spectrum and the presence of hydrogen atoms oscillating with the nanoparticle surfaces which enhances the number of scattered neutrons.

Figure 1

(Color online) $S\left(\omega \right)/S_{\text{el}}$ with experimental error bars for an anatase ${\text{TiO}}_2$ nanopowder with nanoparticles having an average diameter $d=3.6\text{nm}$ (sample HT5). The neutron wavelength is $5.1\text{Å}$ and the temperature is $T=300\text{K}$ before (top) and after (bottom) the heat treatment and atmosphere replacement at 500 K. The black lines obtained by smoothing the data are only guides for the eyes.

Figure 2

(Color online) $S\left(\omega \right)/S_{\text{el}}$ with experimental error bars for two anatase ${\text{TiO}}_2$ nanopowders with nanoparticles having average diameters $d=3.6\text{nm}$ (sample HT5, top) and $d=6.5\text{nm}$ (sample HT7, bottom). The neutron wavelength is $4.1\text{Å}$ and the temperature $T=500\text{K}$.

Figure 3

(Color online) $S\left(\omega \right)$ calculated for sample HT5 with varying Gaussian size distribution. The average diameter is $d=3.6\text{nm}$ in all cases. The FWHMs of the size distribution are 10% and 50% of the average size for the dotted and continuous curves, respectively. The neutron wavelength is $5.1\text{Å}$. The vertical lines at the bottom show the theoretical frequencies of the eigenvibrations for a spherical particle having a 3.6 nm diameter. The height of these lines is proportional to the degeneracy of the modes (either 1 or 2) multiplied by $\text{URS}/{\omega }^3$.