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 2n 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 ͑mx͓͒xh͑1͒͑x͔͒ЈϪh͑1͒͑x͓͒mxj ͑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. Mirrors ͑M3 and M4͒ direct the two beams made far from an object, so that even light that is scattered at through the sample cell ͑SC͒ and the reference cell ͑RC͒, a small angle can be considered removed from the beam. The which are identical 1-cm square quartz cells. The sample cell same paradox arises in quantum mechanical scattering in the is filled with distilled water and suspended particles, and the limit kaӷ1. The total cross section results from the geo- reference cell contains only distilled water. After passing through the cells, the beams continue to matched silicon pho- todiode detectors ͑D1 and D2͒. The beam passing through the sample cell is attenuated due to scattering outside the approximately 3° cone angle subtended by the detector. The dual-beam instrument automatically subtracts out losses in
Fig. 2. A schematic diagram of the spectrophotometer. L is the light source, Fig. 1. Plot of scattering efficiencies, Qϭ/a2, for Rayleigh ͑dashed M1–M4 are mirrors, S1 and S2 are slits, G is the cylindrical grating, BS is curve͒ and Mie ͑solid curve͒ scattering vs mx for mϭ1.59/1.33. The dotted the beam splitter, SC is the sample cell, RC is the reference cell, and D1 and line indicates the limiting value of Qϭ2. D2 are detectors.
621 Am. J. Phys., Vol. 70, No. 6, June 2002 Cox, DeWeerd, and Linden 621 the reference cell due to reflection from the cell walls or absorption or scattering by the distilled water. If the particle number density is sufficiently low, the light will most likely scatter only once in passing through the sample cell. Under these single scattering conditions, the re- duced irradiance I(L) after passing through a sample of length L is related to the initial beam irradiance I0 by I(L) ϭ ϪL I0e . The instrument actually records the optical den- sity, D, of the sample. The optical density is a logarithmic measure of the beam attenuation defined by D ϭ log(I0 /I(L)), so the experimental cross section is given by ϭ͑ln 10͒D/L. ͑3͒ Absorption was negligible for the particles studied so is the scattering cross section. It is necessary that the experiments be performed under conditions for which single scattering is the dominant attenu- ation process. Otherwise, some light scattered out of the Fig. 3. Experimental cross sections ͑points͒ and Rayleigh cross section ͑ ͒ ϭ original beam might be scattered again and reach the detec- solid line vs the wavelength 0 for a 0.0285 m. The error bars repre- sent the experimental uncertainties discussed in the text. tor. This multiply scattered light would be incorrectly re- corded as unscattered, and the final optical density and ex- perimental cross section would be erroneously low. To study the effects of multiple scattering, measurements were made The particle radius of aϭ0.0605 m corresponds to mx ϭ with decreasing number densities corresponding to optical 1.21 at 500 nm. For this mx value, Fig. 1 shows that QRay ϭ densities from D 2.0 to 0.1 at 500-nm wavelength. It was exceeds QMie by 28%. Figure 5 is a graph of experimental, found that as D decreased, the experimental cross section Rayleigh, and Mie cross sections versus 0 . The experimen- rose to a constant maximum value for D below about 0.5. tal cross sections are in good agreement with the Mie theory, Therefore, it was assumed that for D below 0.5, multiple but not with the Rayleigh predictions, as expected. The ex- scattering could be ignored. The experimental cross sections Ϫ4 perimental cross sections are plotted versus 0 for these reported in Sec. IV were obtained from samples with D be- particles in Fig. 6. The curve is seen to still be linear even tween 0.1 and 0.5. though the particle radii are outside the range for the Ray- We consider the question of whether the present experi- leigh theory. However, the experimental slope of (2.38 ment could be performed using other suspensions, such as fat Ϯ ϫ Ϫ41 6 globules from milk. The spectrophotometer could measure 0.03) 10 m does not agree with the value of 2.99 ϫ Ϫ41 6 the wavelength dependence of the attenuation due to the 10 m predicted by the Rayleigh theory. scattering and yield values of vs . These measurements The largest particle radius, aϭ0.2615 m, with mxϭ4.7 would provide a quantitative complement to the common at 500 nm was well outside the range where the Rayleigh classroom demonstration that shows that blue light is scat- theory is valid and should only be compared to the Mie tered more than red by very small particles.19 However, the theory. Figure 7 shows good agreement between the wave- cross sections could not be determined because the number length dependence of the experimental and Mie theory cross density, , would be unknown. Similarly, the theoretical sections. Figure 8 is a plot of the experimental cross section Ϫ4 cross sections for these particles could not be calculated be- versus 0 , which is no longer linear. However, shorter cause the particle radii would also be unknown.
IV. RESULTS AND DISCUSSION Experimental measurements were obtained for the three particle sizes mentioned previously. For each particle size, several experimental samples were prepared from the origi- nal sample supplied by the manufacturer. Repeated measure- ments on these samples produced run to run consistencies within about 2%. In the case of the smallest particle radius, a second sample obtained from the manufacturer also pro- duced similar agreement with earlier experiments. Figure 3 is a plot of experimental total cross sections com- pared with the Rayleigh cross section for a particle radius of aϭ0.0285 m. Even though this radius corresponds to mx ϭ0.57 at 500 nm, there was still good agreement between the Rayleigh theory and the experimental results. Figure 4 Ϫ4 shows the experimental cross sections versus 0 ; the ex- perimental results lie on a straight line with a slope of (3.6 Ϫ43 6 Ϯ0.2)ϫ10 m . This slope compares well with the Ray- Fig. 4. Experimental cross sections vs Ϫ4 for particle radius a Ϫ 0 leigh theory prediction of 3.4ϫ10 43 m6. ϭ0.0285 m with a least-squares straight line fit.
622 Am. J. Phys., Vol. 70, No. 6, June 2002 Cox, DeWeerd, and Linden 622 Fig. 5. Experimental ͑points͒, Rayleigh ͑dashed curve͒, and Mie ͑solid Fig. 7. Experimental cross sections ͑points͒ and Mie cross section ͑solid ͒ ϭ ͒ ϭ curve cross sections vs 0 for a 0.0605 m. curve vs 0 for a 0.2615 m.
three wavelengths. For each particle size, the agreement be- wavelengths are still scattered more than longer ones as ex- tween experiment and theory is within the uncertainty esti- pected for particles smaller than the wavelength. mate. There were several sources of uncertainty in both the ex- There is another interesting consideration with regard to perimental and the theoretical values displayed in the figures. the particle radii used in these calculations. The manufac- The experimental cross sections were calculated from Eq. ͑ ͒ turer specified that a given sample was composed of particles 3 . The uncertainty in the number density, , resulted from ͗ ͘ ͗ ͘ of a certain mean radius, a , with a standard deviation in uncertainties in the average particle radius, a , and in the fraction of particles by volume, f, in the original sample from size uniformity of a . These a values, expressed as a per- cent of ͗a͘, were 2.1%, 4.5%, and 15% for radii of 0.2615, the manufacturer. These values were specified by the manu- facturer to be Ϯ4% for ͗a͘ and Ϯ10% for f. The measured 0.0605, and 0.0285 m, respectively. Hence, the calculated optical density D had an estimated uncertainty of 2%, which cross sections should be calculated as an average over the resulted in an overall uncertainty value of Ϯ15.7% in the appropriate size distribution function. The averaging was experimental cross sections. This uncertainty is represented done analytically for the Rayleigh cross sections. Because 6 by the error bars in Figs. 3–8. Ray is proportional to a , the average effective cross section 6 6 The theoretical Rayleigh cross section is proportional to was found by replacing ͗a͘ by ͗a ͘, the average value of 6 a6, so an uncertainty of Ϯ4% in ͗a͘ results in a 24% uncer- a over the size distribution. If we assume that the distribu- ͗ 6͘ tainty in Ray . Similar uncertainties are inherent in the cross tion function is Gaussian, then a is given by sections calculated from the Mie theory. Table I shows ex- 1 ϩϱ perimental and appropriate theoretical cross sections together ͗ 6͘ϭ ͵ 6 ͓Ϫ͑ Ϫ͗ ͒͘2 2͔ ͑ ͒ a a exp a a /2 a da. 4 ͱ Ϫϱ with the above uncertainties for the three-particle radii at 2 a Equation ͑4͒ yields
Ϫ4 ϭ Fig. 6. Experimental cross sections vs 0 for a 0.0605 m with a least- Ϫ4 ϭ squares straight line fit. Fig. 8. Experimental cross sections vs 0 for a 0.2615 m.
623 Am. J. Phys., Vol. 70, No. 6, June 2002 Cox, DeWeerd, and Linden 623 Table I. Experimental and Rayleigh or Mie cross sections with estimated measurements. These results yield clear confirmation that the uncertainties for three radii. Ϫ4 cross section is proportional to 0 , as is often mentioned in aϭ0.0285 m introductory discussions of atmospheric light scattering. If ͑ ͒ 2 2 studies are done with the larger particles, then the Mie theory 0 nm exp(m ) Rayleigh(m ) is required. The Mie cross sections are easily calculated us- Ϫ18 Ϫ18 450 8.68Ϯ1.22 (ϫ10 ) 8.32Ϯ2.00 (ϫ10 ) ing existing programs, and there is good agreement between Ϯ ϫ Ϫ18 Ϯ ϫ Ϫ18 550 3.75 0.53 ( 10 ) 3.73 0.90 ( 10 ) theory and experiment. 650 1.92Ϯ0.27 (ϫ10Ϫ18) 1.91Ϯ0.46 (ϫ10Ϫ18) ϭ a 0.0605 m APPENDIX: DERIVATION OF THE RAYLEIGH ͑nm͒ (m2) (m2) 0 exp Mie CROSS SECTION 450 5.89Ϯ0.82 (ϫ10Ϫ16) 5.50Ϯ1.32 (ϫ10Ϫ16) 550 2.72Ϯ0.38 (ϫ10Ϫ16) 2.79Ϯ0.67 (ϫ10Ϫ16) Figure 9 shows the electric field of a plane wave in 650 1.41Ϯ0.20 (ϫ10Ϫ16) 1.53Ϯ0.37 (ϫ10Ϫ16) vacuum traveling in the z direction, linearly polarized in the ϭ Ϫ x – z plane given by E(z,t) xˆE0 sin(k0z t), where k0 aϭ0.2615 m ϭ 2 2 2 / 0 . The wave is incident on a dielectric sphere of ͑nm͒ exp(m ) (m ) 0 Mie radius a and real ͑nonabsorbing͒ refractive index n . The Ϫ Ϫ sph 450 3.51Ϯ0.49 (ϫ10 13) 3.47Ϯ0.83 (ϫ10 13) probability that the sphere scatters radiation at angle is Ϫ13 Ϫ13 550 2.30Ϯ0.32 (ϫ10 ) 2.40Ϯ0.58 (ϫ10 ) proportional to the differential scattering cross section, Ϯ ϫ Ϫ13 Ϯ ϫ Ϫ13 650 1.58 0.22 ( 10 ) 1.71 0.41 ( 10 ) d()/d⍀. Integrating d()/d⍀ over all scattering angles yields the total scattering cross section, , which is propor- tional to the probability of scattering in any direction. The ͗a6͘ϭ͗a͘6ϩ15͗a͘42ϩ45͗a͘24ϩ156. ͑5͒ differential cross section is defined as the ratio of the power a a a scattered into the solid angle, d⍀, between and ϩd to Because a /͗a͘ is small, the fractional change in the Ray- the incident power per unit area. The latter is the magnitude leigh cross section due to using ͗a6͘ rather than ͗a͘6 is of the incident time averaged Poynting vector and is given ͗ ͘ 2 ϭ ͗ ͘ ͉͗ ͉͘ϭ 2 approximately 15( a / a ) . For a 0.0285 m, the a / a by Si E0/2 0c. value of 15% should result in a 33% increase in the mea- The scattered power results from the driven oscillating po- sured cross section. However, the experimental results for larization of the dielectric sphere. The primary assumption this particle size were in agreement to better than 5% with involved in Rayleigh scattering is that the sphere diameter is theoretical predictions based on the nominal values of ͗a͘, considerably smaller than the wavelength inside the sphere without making the above correction for this distribution. so that the polarization, P, can be approximated as uniform Our results were consistent with predictions based on a size throughout the sphere. The usual criterion for this assump- ͗ ͘ Ӷ 23 distribution with a / a of 5% or less. Therefore, we specu- tion to be satisfied is nsphk0a 1. Under these conditions late that the width of the particle size distribution might have the entire sphere is considered to be an oscillating dipole of ϭ 3 been smaller than 15%. Wang and Hallett have developed an magnitude P0 P(4 a /3). It is assumed that for visible inversion technique to extract particle size distributions from light the frequency is low enough that resonance absorption 20 extinction spectra. Although their methods are beyond the in the ultraviolet can be neglected. scope of the present study, they might be used to determine The polarization of the sphere is found by solving the the size distribution for these particles. classic problem of a dielectric sphere in a previously uniform The average of the Mie cross sections over the distribution ϭ ⑀ 2 Ϫ 2 ϩ 24 field and is P 3 0E0(nsph 1)/(nsph 2). The radiated of the particle radii cannot be calculated analytically. There- ͑scattered͒ irradiance a distance r from the oscillating dipole fore, the integral was found numerically using Gauss– ͉͗ ͉͘ ϭ 4 2 2 2 2 21 at angle is Ss ʈ ( 0 P0)cos /(32 cr ) for inci- Hermite quadrature. For a function weighted by a Gauss- ͑ ian, the integral may be approximated by dent polarization parallel to the scattering plane the x – z ͒ ͉͗ ͉͘ ϭ 4 2 2 2 plane , and Ss Ќ ( 0 P0)/(32 cr ) for incident po- ϩϱ 25 ͵ Ϫy2 ͑ ͒ Ϸ ͑ ͒ ͑ ͒ larization perpendicular to the scattering plane. For unpo- e f y dy ͚ wn f y n , 6 Ϫϱ n larized incident light the total scattered irradiance is then the
where y n are the roots of the nth order Hermite polynomial 22 and wn are the associated weights. Six terms were suffi- cient to calculate the averaged cross sections to three signifi- ϭ cant figures. For 0 500 nm, the a /͗a͘ percentages given above resulted in increases in the theoretical cross sections of only 2.3% and 0.2% for the radii of 0.2614 and 0.0605 m, respectively. These increases were insignificant compared to the other uncertainties in the theoretical values.
V. CONCLUSIONS We have presented an experiment and analysis that could serve as a convenient introduction to light scattering from Fig. 9. A plane wave polarized in the x – z plane is incident on the sphere small, spherical particles in an undergraduate laboratory. If from the left. Part of the scattered wave is scattered between and only the smallest particle size is studied, then the simple ϩd. The arrows on the sphere indicate the polarization, P, of the dielectric Rayleigh theory gives good agreement with experimental material.
624 Am. J. Phys., Vol. 70, No. 6, June 2002 Cox, DeWeerd, and Linden 624 ͉͗ ͉͘ϭ ͓͉͗ ͉͘ Rohde, ‘‘Laser trapping of microscopic particles for undergraduate experi- average over the two polarizations, Ss 1/2 Ss ʈ ͑ ͒ ͑ ͒ ϩ͉͗ ͉͘ ͔ ments,’’ Am. J. Phys. 68 11 , 993–1001 2000 . Ss Ќ . Multiplying the irradiance by the area subtended 8H. C. van de Hulst, Light Scattering by Small Particles ͑Wiley, New York, 2 by the solid angle d⍀, dAϭr d⍀, yields the total power 1957͒,p.70. scattered into the solid angle. Finally, by using the above 9Craig F. Bohren and Donald R. Huffman, Absorption and Scattering of definition of the differential cross section and substituting the Light by Small Particles ͑Wiley, New York, 1983͒, pp. 82–129. 10 total dipole moment of the sphere, we obtain Reference 3, pp. 288–290. 11Reference 9, pp. 477–482. ͒ 2 Ϫ 2 4 12For example, http://omlc.ogi.edu/software/mie/has links to several sites, d ͑ 1 nsph 1 2 ͯ ϭ ͩ ͪ ͩ ͪ a6͑1ϩcos2 ͒. including the URL, http://omlc.ogi.edu/calc/mie calc.html, where calcula- d⍀ 2 2 ϩ – Ray nsph 2 0 tions can be done interactively online. 13Craig F. Bohren, Clouds in a Glass of Beer: Simple Experiments in Atmo- For a sphere in a surrounding medium such as water with spheric Physics ͑Wiley, New York, 1987͒, pp. 91–97. This book contains refractive index nmed , the index of the sphere is replaced by an amusing anecdote about inadvertently demonstrating this behavior. ϭ 14 the relative index, m nsph /nmed , and the vacuum wave- Reference 8, pp. 107–108. 15Reference 9, pp. 107–111. length, 0 , is replaced by the wavelength in the medium, 16 Richard W. Robinett, Quantum Mechanics: Classical Results, Modern Sys- 0 /nmed . The total scattering cross section is then obtained tems, and Visualized Examples ͑Oxford U.P., New York, 1997͒, pp. 519– by integrating the above equation over the entire solid angle, 520. which yields Eq. ͑1͒. 17Duke Scientific Corporation, 2463 Faber Place, P.O. Box 50005, Palo Alto, CA 94303. a͒ 18Jasco Instruments, Model V530. Electronic mail: Alan–[email protected] 1Andrew T. Young, ‘‘Rayleigh scattering,’’ Phys. Today 35 ͑1͒,2–8͑1982͒. 19Charles L. Adler and James A. Lock, ‘‘A simple demonstration of Mie 2For a collection of research papers on light scattering by the atmosphere, scattering using an overhead projector,’’ Am. J. Phys. 70 ͑1͒, 91–93 see Selected Papers on Scattering in the Atmosphere, edited by Craig ͑2002͒. Bohren ͑SPIE Optical Engineering Press, Bellingham, WA, 1989͒. 20Jianhong Wang and F. Ross Hallett, ‘‘Spherical particle size determination 3Mark P. Silverman, Waves and Grains ͑Princeton U.P., Princeton, NJ, by analytical inversion of the UV-visible-NIR extinction spectrum,’’ Appl. 1998͒, pp. 288–290. Opt. 35, 193–197 ͑1996͒. 4Athanasios Aridgides, Ralph N. Pinnock, and Donald F. Collins, ‘‘Obser- 21S. E. Koonin, Computational Physics ͑Addison–Wesley, New York, 1986͒, vation of Rayleigh scattering and airglow,’’ Am. J. Phys. 44 ͑3͒, 244–247 pp. 83–87. ͑1976͒. 22M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions 5R. M. Drake and J. E. Gordon, ‘‘Mie scattering,’’ Am. J. Phys. 53 ͑10͒, ͑Dover, New York, 1965͒, p. 924. 955–961 ͑1985͒. 23Reference 9, pp. 132–134. 6I. Weiner, M. Rust, and T. D. Donnelly, ‘‘Particle size determination: An 24Roald K. Wangsness, Electromagnetic Fields ͑Wiley, New York, 1986͒, undergraduate lab in Mie scattering,’’ Am. J. Phys. 69 ͑2͒, 129–136 2nd ed., p. 194. ͑2001͒. 25J. D. Jackson, Classical Electrodynamics ͑Wiley, New York, 1975͒,2nd 7Robert Pastel, Akllan Struthers, Ryan Ringle, Jeremy Rogers, and Charles ed., pp. 411–414.
625 Am. J. Phys., Vol. 70, No. 6, June 2002 Cox, DeWeerd, and Linden 625