<p><strong>Based on the Chapter written by Prof. Dr. Jan K. Kruger:<br /></strong></p>
<p><strong>"Brillouin Spectroscopy and its Application to Polymers"</strong></p>
<p><strong>Appeared in the book: Optical Techniques to Characterize Polymer Systems (Edited by Heinz B&auml;ssler) Elsevier 1989</strong></p>
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<p><em>BRIEF INTRODUCTION TO CLASSICAL BRILLOUIN SPECTROSCOPY</em></p>
<p>Brillouin spectroscopy (BS) applied to polymers primarily concerns inelastic light scattering on thermally excited sound waves (phonons) at hypersonic frequencies (~10 MHz to ~100 GHz). The Brillouin scattering effect was predicted first by Brillouin [1] and Mandelshtam [2] and for the first time experimentally verified by Gross [3]. The kinematics of this light scattering process are governed by energy conservation and momentum conservation</p>
<p>&nbsp;h&omega;<sub>s</sub> = h&omega;<sub>i</sub> &plusmn; h&Omega; &nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (1)</p>
<p>&nbsp;<em>h</em><strong>k<sub>s</sub></strong>&nbsp; = <em>h</em><strong>k<sub>i</sub> </strong>&plusmn; <em>h</em><strong>q </strong>&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (2)</p>
<p>&nbsp;&omega;<sub>i</sub> /2&pi;, &omega;<sub>s </sub>/2&pi; and &Omega; /2&pi; = f are the frequencies of the incident laser light, of the scattered light and of the involved sound wave respectively. <strong>k<sub>i</sub></strong> and <strong>k<sub>s</sub></strong> are the wave vectors of the Incident laser light and the scattered light with the sample, and <strong>q</strong> is the wave vector of the sound wave Involved. Within the scattering process phonons of energy h&Omega; are either created (&Omega;<sub>S </sub>= &Omega;<sup>-</sup>, Stokes scattering) or annihilated (&Omega;<sub>AS </sub>= &Omega;<sup>+</sup>, Anti-Stokes scattering). Generally, in solids three orthogonal polarized acoustic modes of different frequencies &Omega; (p, <strong>q</strong>) are related to each wave vector <strong>q</strong> (p = l, 2, 3) A typical BS spectrum of a solid is shown in fig. 1.</p>
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<p style="text-align: center;"><img style="display: block; margin-left: auto; margin-right: auto;" src="upload/images/spectrum.JPG" alt="" width="238" height="324" /></p>
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<td style="text-align: center;"><em>Fig. 1 Brillouin spectrum (of Hexatriacontane at room temperature). The lines denoted by Q<sub>L</sub> and Q<sub>T</sub> represent the quasilongitudinal and quasitransverse acoustic phonons (seen as Stokes [-f] and anti-Stokes [+f] lines) at phonon frequency f. The quantity 2G measures the acoustic losses. R denotes the Rayleigh line which is due to elastic light scattering. For display purposes, the Rayleigh line has been reduced in height by a factor of 300 compared to the phonon lines.</em></td>
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<p>&nbsp;It consists of three pairs of Brillouin lines which are shifted away from the frequency wl of the exciting laser light by the frequency &Omega;(p) . The line width &Gamma;(p, <strong>q</strong>) (half width at half maximum, HWHM) measures the temporal attenuation of the acoustic modes. &Gamma;(p, <strong>q</strong>) is inversely proportional to the life time of the phonons Involved In the scattering process. For moderate sound attenuation (&Gamma;(p, <strong>q</strong>) &lt; &Omega; (p, <strong>q</strong>)) the sound velocity v(p, <strong>q</strong>) is related to the measured frequency &Omega; (p, <strong>q</strong>) by</p>
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<p>v(p, <strong>q</strong>) = &Omega; (p, <strong>q</strong>) / q = f (p, <strong>q</strong>) &Lambda;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; with p = 1, 2, 3 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (3)</p>
<p>&nbsp;where &Lambda; is the hypersonic wavelength.</p>
<p>&nbsp;The elastic stiffness coefficient is related to the sound velocity and the mass density of the material by</p>
<p>&nbsp;c (p, <strong>q</strong>) = &rho; v<sup>2</sup> (p, <strong>q</strong>) &nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (4)</p>
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<p>To account for energy dissipation effects complex quantities are chosen for the elastic stiffness coefficients [c*(p, <strong>q</strong>) = c&rsquo;(p, <strong>q</strong>) - ic&rdquo;(p, <strong>q</strong>)]. c*(p, <strong>q</strong>) Is related to the complex phonon frequency &Omega;*(p, <strong>q</strong>) = &Omega;&rsquo;(p, <strong>q</strong>) - i &Omega;"(p, <strong>q</strong>) = &Omega;&rsquo;(p, <strong>q</strong>) + i 2&Gamma;(p, <strong>q</strong>) by</p>
<p>&nbsp;&Omega;&rsquo;(p, <strong>q</strong>) = &Omega; <sub>0</sub> [c&rsquo;(p, <strong>q</strong>) / c(p, <strong>q</strong>)]<sup>1/2</sup> = [q<sup>2</sup> c&rsquo;(p, <strong>q</strong>) /&rho;]<sup>1/2</sup> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (5a)</p>
<p>&nbsp;with</p>
<p>&nbsp;&Omega; <sub>0</sub>(p, <strong>q</strong>) = q [c(p, <strong>q</strong>) /&rho;]<sup>1/2</sup>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (5b)</p>
<p>&nbsp;and</p>
<p>&Omega;"(p, <strong>q</strong>) = &Gamma;(p, <strong>q</strong>) &asymp; &Omega;&rsquo;(p, <strong>q</strong>) [c&rdquo;(p, <strong>q</strong>) / 2 c&rsquo;(p, <strong>q</strong>)] &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (5c)</p>
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<p>Eq. 5a and eq. 5b have been derived assuming that the elastic properties are independent of frequency. The Influence of dispersion on c&rsquo;(p, <strong>q</strong>) and c"(p, <strong>q</strong>) is often small but can be taken into account in a formal way [4]. It should be noticed that in contrast to ultrasonic experiments where dispersion is reflected by the frequency dependence of the complex stiffness coefficient and in which the wave vector <strong>q</strong>(&Omega;) becomes the complex quantity, dispersion in Brillouin experiments is reflected by the variation of the elastic stiffness coefficient as a function of changes in magnitude of the wave vector. The consequences have been discussed by Evans et al. [5].</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; In isotropic solids the two transversely polarized sound modes degenerate and in neat, non-viscous (isotropic) liquids only the longitudinally polarized sound mode can propagate. For isotropic and cubic materials the following relation holds between the magnitude of the acoustic wave vector q = |<strong>q</strong>| and the scattering angle &Phi; ^ (k<sub>i</sub>, k<sub>s</sub>)</p>
<p>&nbsp;q = 2 k<sub>i </sub>sin(&Phi; /2) = (4&pi;n/&lambda;<sub>0</sub>) sin(&Phi; /2)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (6)</p>
<p>where &lambda;<sub>0</sub>is the wave length of the laser light and n is the refractive index of the sample. The case of less symmetric becomes more complicated and is treated in section II.2 of the chapter.</p>
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<h1><strong>References</strong></h1>
<p>[1]&nbsp;&nbsp;&nbsp;&nbsp; L. Brillouin, Ann. De Physique, 17, 88 (1922)</p>
<p>[2]&nbsp;&nbsp;&nbsp;&nbsp; L. I. Mandelshtam, Zh. Russ. Fiz. Khim. Obs., 58, 381 (1926)</p>
<p>[3]&nbsp;&nbsp;&nbsp;&nbsp; E. Gross, Nature, 126, 201,400,630 (1930)</p>
<p>[4]&nbsp;&nbsp;&nbsp;&nbsp; C. J. Montrose, V. A. Solovyev, T. A. Litivitz, J. Acoust. Soc. Am., 43,117 (1968)</p>
<p>[5]&nbsp;&nbsp;&nbsp;&nbsp; W. A. B. Evans, J. G. Powels. J. Phys. A, 7, 1944 (1974)</p>
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