Verification of the Rayleigh scattering cross sectionSayan Chakraborti

Citation: American Journal of Physics 75, 824 (2007); doi: 10.1119/1.2752825 View online: http://dx.doi.org/10.1119/1.2752825 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/75/9?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in Rayleigh scattering revisited: From gases to crystals Am. J. Phys. 78, 94 (2010); 10.1119/1.3236688 Approximation for the Rayleigh Resolution of a Circular Aperture Phys. Teach. 47, 288 (2009); 10.1119/1.3116839 Phonon scattering cross section of polydispersed spherical nanoparticles J. Appl. Phys. 99, 084306 (2006); 10.1063/1.2188251 Thermal transport cross section and phase function of longitudinal phonons for scattering by nanoparticles andmicroparticles J. Appl. Phys. 96, 5202 (2004); 10.1063/1.1789633 An experiment to measure Mie and Rayleigh total scattering cross sections Am. J. Phys. 70, 620 (2002); 10.1119/1.1466815

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Verification of the Rayleigh scattering cross sectionSayan Chakrabortia�

Tata Institute of Fundamental Research, Mumbai, India

�Received 12 February 2007; accepted 22 May 2007�

A simple experiment is described for the direct determination of the wavelength dependence of theRayleigh scattering cross section using the classic example of the blue sky. Suggestions forinclusion into an undergraduate lab are discussed. © 2007 American Association of Physics Teachers.

�DOI: 10.1119/1.2752825�

I. INTRODUCTION

Rayleigh scattering1 is the process of scattering of light byparticles much smaller than the wavelength of the incidentelectromagnetic radiation and is the cause of the blueness ofthe sky. In this simple experiment the wavelength depen-dence of the Rayleigh scattering cross section is determinedusing a commercially available spectrometer.

II. THEORY

Following Jackson2 we can treat the scattering of radiationof angular frequency � by a single nonrelativistic particle ofmass m and charge e under a spherically symmetric, linearrestoring force m�0

2x. From the definition of the differentialcross section, the cross section for scattered light of fre-quency � and polarization � is

d���,�,���d�

= � r� · Erad

E0�2

= � e2

mc2�2

�� · ���2� �4

��02 − �2�2 + �2�2 , �1�

where r is the distance from the scatterer, �� is the unitvector representing the direction of r, and � is the dampingcoefficient. In the large wavelength limit, the scattering crosssection reduces to

d���,��d�

= � e2

mc2�2

�� · ���2� �

�0�4

. �2�

Advanced undergraduate students are referred to the el-egant treatment in Ref. 2. For a heuristic derivation, we canmodel the scatterers as simple harmonic oscillators. Theequation of motion is

m�x + �02x� = F�t� , �3�

where the force due to the electric field is F�t�= �e /m��E0e−i�t. The steady state solution is

x =eE0e−i�t

m��02 − �2�

� . �4�

The dipole moment is proportional to xv and the charge. Theradiated power is proportional to the square of the amplitudeof the dipole oscillations. Thus, for small �0, the scattering

−4

cross section is proportional to � at large wavelengths.

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III. EXPERIMENT

An AvaSpec-2048 Fiber Optic spectrometer was used torecord spectra. The data were averaged over ten readings andsuitably dark subtracted. Due to the finite temperature of thedetector, there is a nonzero current. Readings were takenwith the shutter closed to evaluate this current and later sub-tracted from the final data. In the region of interest, 400 to700 nm, only counts above 1000 were considered. Becausethe data were averaged over ten readings, the data representsa photon count of more than 10 000. Assuming randomnoise, the fractional error at each wavelength is less than 1%.

The idea is to compare the spectrum of the blue sky withthat of the solar spectrum to determine the scattering crosssection at each wavelength. Direct solar spectra could not betaken because the spectrometer was saturated even at thelowest allowed exposure of 2 ms. The lunar spectra, which isvery similar to the solar spectra, was used instead.

The sky spectra is bluer than the lunar spectra. This blue-ness of the sky is supposedly due to Rayleigh scattering bythe atmosphere. We shall now try to extract its wavelengthdependence and see if it matches with theory.

IV. DATA ANALYSIS

Care must be taken to analyze what each count represents.The counts in the lunar spectra can be represented as

M��� = C1����AL���US��� , �5�

where C1 is an undetermined constant, and �, AL, and US arethe efficiency of the detector, the lunar reflectivity, and therelative energy density of solar spectrum, respectively, all atwavelength �. The counts in the sky spectra are

S��� = C2����Rcs���US��� , �6�

where Rcs is the Rayleigh scattering cross section. We wishto determine its wavelength dependence from our data. Theefficiency of the detector and the solar spectra can be elimi-nated by taking the ratio of the two spectra. The lunar albedois approximately constant over the range 400 to 700 nm. Be-cause the Moon lacks an atmosphere, the average particlesize of the scatterers on the lunar surface is much larger thanthe incident light wavelength.

The spectra are shown in Fig. 1. We take the logarithmbase 10 of both spectra and plot their difference against the

logarithm of the wavelength in Fig. 2. The best fit straight

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line is found to have a slope of −3.8. The theoretical curvewith a slope of −4.0 is plotted for comparison.

The experiment can be completed by taking a lunar spec-tra one night and the sky spectra the next morning. If lunarobservations are not possible due to time constraints, a neu-tral density filter may be used to reduce the solar radiation towithin the instrumental limits. Filter response curves, unlesscompletely flat, should be provided by the instructor. Other-wise, both the sky and solar spectra should be recordedthrough the same filter to cancel its effect on the final slope.Sunlight reflected from a white sheet of paper, instead of themoon, has been used to reproduce a similar result.

Fig. 1. Spectra of the moon and sky. Note that the pea

Fig. 2. Relative scattering cross section. Note the almost linear nature of t

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V. POLARIZATION STUDY

Further additions to our experiment might include a polar-ization study of the scattered light. A polarizing filter can beplaced in front of the fiber inlets to let in light of one polar-ization. Spectra should be recorded for different positions ofthe filter and the relative cross sections and plotted as a func-tion of the orientation angle for a given wavelength. Studentscan fit straight lines to the different sets of relative scatteringcross sections to check that they have similar slopes butdifferent intercepts.

shifted toward shorter wavelengths in the sky spectra.

he data points in a log-log plot, indicating a power law dependence.

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ACKNOWLEDGMENTS

The author would like to thank Professor D. K. Ojha andProfessor A. K. Ray for their guidance, my classmate TarakThakore, the Infrared Astronomy Group at TIFR for helpwith observations, and Mehuli Mondal for discussionson Rayleigh scattering. Thanks are also due to the two

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anonymous referees for their constructive reviews and appro-priate suggestions.

a�Electronic mail: sayan@tifr.res.in1Lord Rayleigh �J. W. Strutt�, “On the transmission of light through anatmosphere containing small particles in suspension, and on the origin ofthe blue of the sky,” Philos. Mag. 47, 375–384 �1899�.

2J. D. Jackson, Classical Electrodynamics, 3rd ed. �Wiley, New York,

1998�, Chap. 10.

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