Brillouin and Raman scattering in natural and isotopically controlled diamond

Abstract

The effects of zero-point motion and the anharmonicity of the lattice vibrations of diamond have been explored theoretically in the context of a valence force model explicitly incorporating the isotopic composition. The predictions are tested in a study of the elastic moduli (c i j ) deduced from Brillouin spectra and the zone center optical mode frequency ( 0 ) from Raman spectra of isotopically controlled diamond specimens. On the basis of the anharmonicity parameter of the model associated with bond stretching, deduced from a comparison of the theory with experimentally reported dependence of the lattice parameter with the atomic fraction of 13 C in 12 C 1Ϫx 13 C x diamond, it is predicted that the bulk modulus of 13 C diamond exceeds that for 12 C diamond by one part in a thousand, just below the experimental sensitivity accessible with Brillouin measurements; 0 exceeds the value expected from the M Ϫ1/2 dependence, where M is the average atomic mass, by ϳ 0.3 cm Ϫ1 , consistent with observation. The Grüneisen parameter for 0 and the third-order bulk modulus are consistent with the theoretical estimates from the present model. The elastic moduli for natural diamond determined in the present study, viz., c 11 ϭ10.804(5), c 12 ϭ1.270(10), and c 44 ϭ5.766(5) in units of 10 12 ͑dyn/cm 2 ) are the most accurate yet obtained. ͓S0163-1829͑96͒03330-9͔

Isotope splitting of the zero-phonon line of Fe2+ in cubic III–V semiconductors

“…Work on the effects of isotopic composition includes the discovery of the enhancement of the thermal conductivity of isotopically enriched Ge relative to that of natural Ge [3], the measurement of isotopic effects on the lattice parameter of Ge [4] and diamond [5] and the dependence of the frequency of the zone center F 2g (Γ 25 + ) optical mode on the inverse square-root of the mass observed in the Raman spectrum of monoisotopic Ge [6,7] and diamond [8]. More recently, the effects of the anharmonicity of the lattice vibrations and of the zero-point motion of the atoms have been observed and analyzed in natural and 13 C-enriched diamond [9,10]. Reviews of some aspects of these topics have been given by Ramdas [11] and Haller [12].…”

Basic Properties of Diamond: Phonon Spectra, Thermal Properties, Band Structure

“…Hence the speed of the phonon (the speed of that sound wave) is known, and the speed of sound is linked to the elastic constants. The values at room temperature, for natural isotope abundance diamond, have been measured by Brillouin scattering [11]: c 11 = 1080.4 ± 0.5, c 12 = 127.0 ± 0.7, c 44 = 576.6 ± 0.7 GPa.…”

“…The stress, s, required to produce this particular deformation is given by the force acting in the x direction, on an area whose normal is in the x direction, and so is denoted s xx . The relationship between the stress and the strains is given in terms of the elastic constants c 11 A different type of distortion is a shear, for example, when a cube face normal to y is sheared in the z direction to give a strain e yz . To achieve this strain requires a shear stress, which is a force acting in the z direction on a unit-area face whose normal is in the y direction:…”

“…The two-phonon Raman scattering spectrum of diamond is well known [11,20] and has been fitted closely, using five parameters to describe the second-order polarisability [22]. By definition, the multi-phonon processes involve combinations of (usually different) phonons.…”

High quality factor indium oxide mechanical microresonators

“…Therefore, it was estimated by calculating the total mass of the rods from the indium oxide density and by using the same approximation for the cross section employed before. Thus, 30 compared to diamond ($1200 GPa) 39 or other materials such as SiC (E $ 420 GPa) 40 or Si 3 N 4 ($280 GPa), 41 which combined with the high Q, leads to a high sensitivity. Since the stiffness of a rectangular cross section rod increases with the third power of its thickness, it should be possible to further reduce the spring constant of the rods by decreasing their thickness, thus reducing their F min even more.…”