Raman Spectroscopy of nanomaterials - institut de chimie et des ...

(a)ARTICLE IN PRESS+ MODEL18 G. Gouadec, Ph. Colomban / Progress in Crystal Growth and Characterization of Materialsxx (2007) 1e56(b)e rezMθeϕeθrxOy(c)Fig. 7. (a) Description of one nanoparticle in the Elastic Sphere Model (ESM). (b) Oscillation of a free sphere in a singlespheroidal mode. (c) Oscillation of a free sphere in a single torsional mode.~u 2 ¼ lðl þ 1Þ j lðk L rÞk L r ~e jl ðk L rÞlmðq; 4Þþ þ j 0 lk L rðk LrÞ ~n lm ðq; 4Þ ð25Þ~u 3 ¼ j l ðk T rÞ~m lm ðq; 4Þ ð26ÞA distinction is possible between two families of modes (see Fig. 7b and c):Modes with purely radial displacement are called spheroidal modes.Modes with purely tangential displacement, occurring with no volume change, are calledtorsional modes.Duval [259] was the first to establish the selection rules for the ESM. Since the modes irreduciblerepresentations had to coincide with those of the sphere group, he showed that only purelyspherical (l ¼ 0; irreducible representation D g (0) ) and quadrupolar (l ¼ 2; D g (2) ) spheroidal modesare Raman active (while all torsional modes are forbidden). 13 Their relative intensity is variablebut only the first order ‘‘surface’’ ones (n ¼ 0), those shown in Fig. 8, have significant contributions[257,260e262]. Their identification is easy since only D g (0) is polarized (parallel polarization).An expression for the stress field can be derived from Eqs. (24)e(26) and the subsequent13 ( lIn group theory, the D ) i notation indicates modes of (2l þ 1) degeneracy with either symmetry (i ¼ g) or antisymmetry(i ¼ u) with respect to an inversion centre.Please cite this article in press as: G. Gouadec, Ph. Colomban, Prog. Cryst. Growth Charact. Mater. (2007),doi:10.1016/j.pcrysgrow.2007.01.001