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ARTICLE IN PRESS+ MODEL12 G. Gouadec, Ph. Colomban / Progress in Crystal Growth and Characterization of Materialsxx (2007) 1e56Intensity / a.u.Rayleigh wingAmorphous matter+ "oriented amorphousmoities"Crystallinephase//0 100 200 0 500 1000 1500Wavenumber / cm -1Fig. 5. Low wavenumber Raman spectra of the PA 66 polyamide fibre (FUHP grade, Rhodia) recorded with the excitingelectric field polarized either parallel (//) or perpendicular (t) to the fibre axis (l exc ¼ 514.5 nm) [171]. The collectivemode at w100 cm 1 is highly polarized. The zoom shows how the crystalline and amorphous (much wider) contributionscan be analysed separately [169].3.3. Size determination in nanomaterialsTwo models are widely used to derive particle size from Raman spectra. The Phonon ConfinementModel (PCM) projects Raman ‘‘inactive’’ bulk modes onto the BZ c whereas the ElasticSphere Model (ESM) describes the free oscillations of homogeneous spheres.3.3.1. The Phonon Confinement Model (PCM)Richter et al. [179] proposed a very intuitive Phonon Confinement Model (PCM) for thephonons in nanospheres of diameter L. They simply multiplied the plane wave describing a phonon,with wavevector ~k 0 (in a perfect crystal Fð~k 0 ;~rÞ ¼uð~k 0 ;~rÞe i ~k 0 $~r , u having the same spatialperiodicity as the lattice) by a Gaussian function:Fð~k 0 ;~rÞ¼e aðr LÞ 2 uð~k 0 ;~rÞe i ~k 0 $~rð9ÞAssuming uð~k 0 ;~rÞwuð~rÞ and using FT to refer to Fourier Transforms, Eq. (9) is equivalentto:i ZFð~k 0 ;~rÞfFThFT1 e aðr LÞ 2 e i ~k 0 $~r¼d 3 ~k Cð~k 0 ;~kÞ$e i ~k$~rð10Þwith Cð~k 0 ;~kÞ¼ 1ð2pÞ 3 Zd 3 ~r e aðr LÞ 2 e ið ~k 0~kÞ~rð11ÞThus, the wave associated with a phonon confined in an imperfect crystal simply is a superpositionof plane waves with jCð~k 0 ;~kÞj 2 weight. Recalling that each wave gives rise to aLorentzian (Eq. (4)), the total Raman intensity is eventually given by: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
ARTICLE IN PRESS+ MODELG. Gouadec, Ph. Colomban / Progress in Crystal Growth and Characterization of Materialsxx (2007) 1e56ZIðnÞa d 3 ~kBZ½njCð~k 0 ;~kÞj 2nð~kÞŠ 2 þ 2 G0213ð12ÞEq. (12) mathematically expresses the Raman selection rule breaking induced by phonon confinement8 with a weighed exploration of the dispersion curves. Taking k BZe astheedgeofBZ,q as thereduced wavevector (q ¼ k/k BZe ) and assuming isotropic mode dispersion [180], Eq.(12) yields:IðnÞaZ q¼1q¼0dqe k2 BZe ðq q 0Þ 2 L 22a½n1nðqÞŠ 2 þ 2 G02ð13ÞThis is the equation to which most authors refer when using the PCM [179e186]. 9 For semiconductorQuantum Dots (QDs), NanoWires (NWs) or slabs, the PCM is easily adapted usingthe appropriate expressions for the d 3 ~k integration volume in Eq. (12) [189e193]. 10 Knowledgeof the Vibrational Density of States (VDOS) is required for computing Eq. (13). It may be obtainedeither from neutron scattering measurements, from data on parent structures or from abinitio calculations based on a rigid-model structure [182,183,195e197]. The PCM, which doesnot apply to the acoustic modes because their energy is nil at BZc is very seldom used for theTO modes on account of their low dispersion [192,198]. It is almost exclusively applied to theLO modes. With LO wavenumbers usually being maximum at BZc, the integration in Eq. (13)introduces additional contributions on the low frequency side of the single crystal mode and theresulting peaks become asymmetric. Peak adjustment is not always mandatory once phononconfinement has been invoked. In first approximation, the peak shift is indeed proportionalto the inverse of the grain size [187,199]. The overall half-width at half-height also is, proportionalto the inverse of the grain size, as was reported for nanocrystalline CeO 2 [196,197] (seeFig. 6) or boron nitride [187] and can be verified with data on Ge [184].In the paper introducing the PCM, Richter et al. [179] assumed a ¼ 2. Campbell and Fauchet[193] later tested several forms for the weighing function introduced in Eq. (9). They came to8 As a consequence of Heisenberg’s principle, the uncertainty Dk ¼ Dp/Z on the wavevector must remain above orequal to (2L) 1 .9 Based on the triple hypothesis that modes from different crystallites are uncorrelated, that lifetimes can be simulatedwith a Lorentzian broadening (G 0 ) and that susceptibility variations (Dc) are proportional to the normal vibration coordinates,Nemanich et al. [187] predicted the Raman intensity on probing of N orthorhombic crystallites to be:IðnÞaSð~k 0 ; nÞ¼N n B ðnÞþ1V X 1 C ~k; nc 2 V 2 j ð~kÞ jFð~k ~k 0 Þj 2 G 0 =4p n~k;jn nj ð~kÞ 2 ðN3Þþ G 2 0=4In this equation, S is the Fourier transform of polarisability variations associated with Raman scattering, ðn B ðnÞþ1Þ isthe Bose population factor ðn B ðnÞ ¼1=ðe hn=kT 1ÞÞ, C is the Raman coupling coefficient for branch j and jF(k k 0 )j 2 isthe uncertainty on the wavevector induced by phonon confinement [187,188]. This approach is entirely equivalent to thePCM if the F function is assumed to be Gaussian and the occupation term is neglected (this simplification is justifiedonly when a single band is modelled).10 Note that the PCM failed to predict the Raman shift in silicon spherical or columnar nanocrystals. Zi et al. [194]showed with a Bond Polarisability Modelling that it could rather be predicted using the following expression: a gn vib ¼ n bulk AðN4ÞLIn Eq. (N4), a is the lattice constant whereas A and g fully characterize the nanocrystal geometry.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
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