An experiment to measure Mie and Rayleigh total scattering cross sections A. J. Cox, Alan J. DeWeerd,a) and Jennifer Linden Department of Physics, University of Redlands, Redlands, California 92373 ͑Received 9 July 2001; accepted 7 February 2002͒ We present an undergraduate-level experiment using a conventional absorption spectrophotometer to measure the wavelength dependence of light scattering from small dielectric spheres suspended in water. The experiment yielded total scattering cross-section values throughout the visible region that were in good agreement with theoretical values predicted by the Rayleigh and Mie theories. © 2002 American Association of Physics Teachers. ͓DOI: 10.1119/1.1466815͔ I. INTRODUCTION II. THEORY One of the most important examples of interaction at the The Rayleigh cross section is valid for spherical particles microscopic scale is the phenomenon of scattering. For ex- that have radii small compared to the wavelength of the scat- ample, much of what has been learned about the structure of tered light. Its derivation involves important concepts such as the nucleus, indeed even its discovery, was the result of scat- the definition of the scattering cross section, the polarization tering experiments. Similarly, the analysis of scattering has of a dielectric sphere, and radiation from a driven oscillating yielded most of our present knowledge of elementary par- dipole. Because the derivation should be understood by stu- dents who perform this experiment, it is included as an ap- ticle physics. Compton scattering of x rays by electrons is pendix. For the Rayleigh approximation the cross section can often cited as experimental evidence for the particle nature of be written as the analytic function, the photon. One of the earliest examples of scattering to be studied was that of light scattered by the atmosphere, which 8␲ 2␲n 4 m2Ϫ1 2 ␴ ϭ ͩ medͪ 6ͩ ͪ ͑ ͒ was studied by Tyndall, Rayleigh, and others at the end of Ray ␭ a 2ϩ , 1 3 0 m 2 the nineteenth century.1 The wavelength dependence of scat- ␭ tering by the atmosphere is responsible for both the blue sky where 0 is the vacuum wavelength, a is the particle radius, 2 ϭ and red sunset. Light scattering is of such great importance and m nsph /nmed is the ratio of the refractive index of the in optics that Mark P. Silverman wrote that, ‘‘virtually every particle to that of the surrounding medium. The Rayleigh 3 ␭Ϫ4 aspect of physical optics is an example of light scattering.’’ cross section is clearly proportional to 0 . Several light scattering experiments useful for teaching The Mie cross section, which is valid for spheres of any undergraduates have appeared in this journal. Aridgide, Pin- size, is obtained from a considerably more complicated cal- nock, and Collins measured the spectrum of light scattered culation than the Rayleigh approximation. The details of its by the atmosphere.4 Their measurements yielded the ex- derivation are discussed in Refs. 8 and 9, and summaries of pected wavelength dependence of the spectrum of scattered the calculation are in Refs. 6 and 10. The solution involves light, but not absolute scattering cross sections. Experiments an incident plane wave and an outgoing spherical scattered by Drake and Gordon5 and recent work by Weiner, Rust, and wave. Because of the spherical symmetry of the system, the Donnelly,6 measured the angular distribution of light scat- incident wave is expanded as an infinite series of vector tered from small particles at a single wavelength. They were spherical harmonics. The components of the total ͑incident able to fit their data to Mie theory calculations and accurately plus scattered͒ electric and magnetic fields tangent to the determine particle radii. Pastel et al.7 performed similar an- surface of the sphere are required to be continuous across the gular measurements with single liquid drops to determine boundary. After considerable mathematical manipulation, the their size. scattered fields are determined and from these fields, the dif- This paper discusses an experiment that measures total ferential and total cross sections are found. The Mie total light scattering cross sections for small particles throughout scattering cross section is expressed as the infinite series ϱ the visible spectrum using a conventional absorption spectro- 2␲ photometer. Because this instrument is available in most ␴ ϭͩ ͪ ͚ ͑ ϩ ͉͒͑ ͉2ϩ͉ ͉2͒ ͑ ͒ Mie 2 2n 1 an bn , 2 chemistry departments, no special apparatus need be con- kmed nϭ1 structed. With this instrument absolute total cross sections ϭ ␲ ␭ where kmed 2 nmed / 0 . The coefficients an and bn are can be easily obtained by undergraduate students in one given by laboratory period. The experimental cross sections agreed ␮ 2 ͒ ͒ Ϫ␮ ͒ ͒ well with the exact Mie theory calculations, which were per- m jn͑mx ͓xjn͑x ͔Ј 1 jn͑x ͓mxjn͑mx ͔Ј a ϭ , formed with an existing computer program. The smallest par- n ␮ 2 ͒ ͑1͒ ͒ Ϫ␮ ͑1͒ ͒ ͒ m jn͑mx ͓xh ͑x ͔Ј 1h ͑x ͓mxjn͑mx ͔Ј ticles studied were sufficiently small so that their measured n n ␮ ͒ ͒ Ϫ␮ ͒ ͒ cross sections also agreed well with those calculated using 1 jn͑mx ͓xjn͑x ͔Ј jn͑x ͓mxjn͑mx ͔Ј b ϭ , the Rayleigh approximation. The concepts of scattering cross n ␮ j ͑mx͓͒xh͑1͒͑x͔͒ЈϪ␮h͑1͒͑x͓͒mxj ͑mx͔͒Ј section and beam attenuation discussed in this paper may be 1 n n n n extended to other types of scattering ͑for example, Compton where the jn’s are spherical Bessel functions of the first kind, ␮ ␮ and Rutherford scattering͒ prominent in the undergraduate the hn’s are spherical Hankel functions, and 1 and are the curriculum. magnetic permeability of the sphere and surrounding me- 620 Am. J. Phys. 70 ͑6͒, June 2002 http://ojps.aip.org/ajp/ © 2002 American Association of Physics Teachers 620 ␮ ϭ␮ ͑ ͒ ␲ 2 dium, respectively. For the present case 1 , and hence metrical contribution classical limit of a and an equal ϭ ␲ ␭ they cancel. The quantity x (2 nmeda)/ 0 is called the size contribution from the diffraction of the incident plane wave parameter and primes indicate derivatives with respect to x. at the sharp edge of the sphere. The diffraction contribution ␴ Numerical values of Mie were calculated using the subrou- is strongly peaked forward around a scattering angle of 11 ␪ Ϸ tine BHMIE. This computer code and similar ones are max 1/ka. For nearby macroscopic objects this diffracted 12 ␴ ␪ϭ readily available online. The series expression for Mie light is not distinguishable from unscattered light at 0, converges after a number of terms slightly larger than the and the paradox is not observed.16 size parameter. For example, the largest particles studied, a ϭ0.2615 ␮m, required only five terms to converge to four significant figures for xϭ2.96. III. EXPERIMENT The commonly used criterion for the validity of the Ray- The experiment consists of measuring the attenuation of leigh approximation is that mxӶ1. To compare the behavior ␴ ␴ an unpolarized light beam as it passes through a sample of of Ray and Mie , calculations were performed over a range spherical particles suspended in water. The polystyrene of values of mx. Figure 1 presents the results in terms of the spheres with refractive index n ϭ1.59 were obtained from respective scattering efficiencies, Qϭ␴/␲a2, versus mx. For sph Duke Scientific Corporation.17 Particles with radii of a the smallest particles, aϭ0.0285 ␮m, with mxϭ0.57 at 500 ϭ0.0285, 0.0605, and 0.2615 ␮m were studied. Samples of nm, the Rayleigh and Mie cross sections differ by only 5.4%. ␳ Ӷ various number densities, , were prepared based on the Thus even though the requirement of mx 1 is not strictly manufacturer’s specification that the purchased samples con- satisfied for our smallest particles, the Rayleigh cross section sisted of 10% particles by volume in water. As recommended is still a useful approximation for them. Figure 1 also dem- by the manufacturer, the original sample container was onstrates how drastically the predictions of the two theories placed in an ultrasonic bath for a few minutes to gently stir differ for size parameters much larger than unity. For ex- and distribute the particles evenly throughout the water. Then ample, Mie theory predicts that the efficiency does not al- a measured amount of particles and water were added to ways rise as the size parameter increases, which means that distilled water to obtain suitable number densities. for certain particle sizes the cross section will actually be- The instrument used to measure the attenuation of light as come larger with increasing wavelength.13 This behavior is Ϫ a function of the wavelength was an absorption ␭ 4 18 in sharp contrast to the 0 wavelength dependence of the spectrophotometer. Before performing the experiment, stu- Rayleigh cross section. dents are asked to remove the outside cover of the instrument For large x ͑that is, particle radius much larger than the and compare the exposed optical components to the sche- wavelength͒, it might be expected from geometrical optics matic diagram in Fig. 2. Light from a halogen lamp ͑L͒ is that the total cross section from the exact calculation ͑Mie reflected and focused by a cylindrical mirror ͑M1͒ onto a slit theory͒ would be ␲a2, and thus Q would approach 1. How- ͑S1͒. After passing through the slit, the expanding beam is ever, Fig. 1 shows the interesting and somewhat puzzling diffracted and refocused by a cylindrical grating ͑G͒ onto trend that as x becomes large, the scattering efficiency ap- another slit ͑S2͒. The quasimonochromatic light from S2 proaches 2. This phenomenon is called the ‘‘extinction para- with a spectral bandwidth of 2 nm is then collimated by a dox’’ and is discussed in detail by van de Hulst14 and Bohren spherical mirror ͑M2͒ and divided into two beams by a beam and Huffman.15 The result is apparent only for observations splitter ͑BS͒.