Genetic Algorithm Inversion of Dual Polarization Polar Nephelometer Data to Determine Aerosol Refractive Index

Abstract

A polar nephelometer that measures the light scattered into a two dimensional plane by a stream of aerosols that intersect a 350 milliwatt diode laser beam with a wavelength of 670 nm is described. A half wave plate is used to orient the laser light both parallel and perpendicular to the measurement plane. This scattering information combined with simultaneously measured size distribution information is used to determine the real (m_r) refractive index of the aerosols using a Genetic Algorithm search method. Errors in the retrieved m_r based on noise in the scattering measurement as well as uncertainties in the size distribution measurement are examined. The m_r of polystyrene latex spheres and ammonium sulfate droplets are determined and match expectations within experimental uncertainties using Mie-Lorenz theory. The angular scattering properties of spherical secondary organic aerosols generated by oxidizing α -pinene do not follow this theory and calls into question the inferred m_r.

INTRODUCTION

Aerosols have been identified as a major uncertainty in predicting the global energy budget (IPCC 2001) because of major gaps in our knowledge of the composition and optical properties of these particles and their direct and indirect effects. Measured and inferred refractive indices for several types of black carbon aerosols (BC) range from 1.1 to 2.75 in the real part (m_r) and from 0 to 1.46 in the imaginary part (m _i ) (Seinfeld and Pandis 1998; Horvath 1993; Fuller et al. 1999; Bond and Bergstrom 2006).Many of these uncertainties arise because the optical properties of these particles depend on the composition and physical form of the particles. For instance the refractive index of bulk carbon is well known, however, while the fundamental absorption and scattering properties of black carbon particulate material may be fairly constant (although not well known, e.g., Bond and Bergstrom (2006)), the refractive index of non-spherical BC in small particulate agglomerates is highly variable. Further, black carbon rarely exists in pure form in the atmosphere; instead it is often coated or combined with other materials in particles, and exists as part of an external mixture of different types of particles. For example it is believed that BC material that is incorporated in a non-absorbing component (i.e., water or organics) can have enhanced absorptive properties (Jacobson 2000; Chylek and Hallett 1992). Finally, BC optical parameters can change in time and as a function of the environment. The refractive indices are a basic optical property of aerosol particles, which are used, often via Mie-Lorenz theory, to determine the parameters relevant to radiative transfer, i.e., the single scattering albedo, asymmetry factor and specific absorption. Clearly, a means to reliably determine realistic aerosol refractive indices is required.

Refractive index data is available for a limited number of substances common in the atmosphere but in pure form. Refractive index data for mixed aerosols is almost non-existent. Refractive indices of external mixtures are linear combinations of the “pure” particles but refractive indices of internal mixtures are not simply derived. Key to the Mie calculation is the combined refractive index of the aerosol mixture, however, the refractive index for internal mixtures is difficult to determine because the degree of mixing needs to be known, which constituents are mixed, and where the inclusion (if any) is located within the particle. And because most aerosols are not spheres, the Mie solution is insufficient in modeling the radiative effects of aerosols (Mishchenko et al. 1995).

There are several means to measure the optical properties of aerosols collected onto a filter substrate, but these are prone to systematic errors because the particles are tightly packed together, interact with the filter material and volatile components in the aerosol dissipate during the necessarily long collection periods (Hänel 1988; Horvath 1997). Our instrument examines the optical properties of the particles as a suspended aerosol and in real time. Thus we can examine, in detail, the single scattering properties and refractive index of the particles as they evolve chemically and physically in their environment. Because we measure the scattering properties, we can also estimate the asymmetry factor by extrapolating the scattering intensities in the forward and reverse direction. The angular scattering intensities, which we measure directly, is in and of itself a very useful parameter. For example, it could provide much needed insights into the source of disagreement between modeled aerosol optical properties and Mie calculated–derived optical properties in closure studies.

There are other methods used to determine the refractive indices of aerosols in the laboratory (Toon and Pollack 1976), but as the instrument presented in this article is based on measurements of the light scattering properties of suspended particles, only similar instruments are reviewed. Goniometer type nephelometers employ a telescope-detector sensor that rotates about the scattering center and have the advantage of allowing determination of the polarization properties of the scattered light. Tanaka et al. (1983) measured the parallel and perpendicular components of scattered light and a library or table lookup search method to recursively determine the real and refractive indices and size distribution, while Zhao et al. (1997), and Zhao (1999) used an inversion of the Mie-Lorenz scattering theory and a measurement of the full Stokes parameters to determine these properties. However, the goniometer type instruments are necessarily complex and large and thus usually confined to laboratory studies. Multi-channel polar nephelometers measure the scattering intensities at several angles simultaneously and have the advantage of measurement speed, and fewer moving parts, and thus can be used in the field. Jones et al. (1994a, 1994b) inverted the absolute intensity measurements of a 15 channel polar nephelometer to retrieve both the real refractive index and the particle size distribution. The DAWN-A light scattering instrument (Wyatt et al. 1988) includes scattering measurements at six polar positions to determine the real refractive index of spherical aerosols (Dick et al. 1998). Via the spherical assumption, the ratio of particle forward to backwards scattering into a large angular swath (i.e., greater than 10°) also contains information about the optical properties of the scattering particle as demonstrated by Szymanski et al. (2002) who utilizes two distinct laser wavelengths to determine the complex refractive index while Baumgardner et al. (1996) uses a similar instrument design but a single wavelength (780 nm) to determines the real refractive index. The instrument design described in this paper combines the advantages of the multi-channel polar nephelometer with the polarization measurement capabilities of the goniometer nephelometer by modulating the polarization state of the incident laser light.

Genetic algorithms (GA) have been used to determine the aerosol size distribution from measurements of the single scattering properties of aerosols (Ye et al. 1999; Lienert et al. 2003), but these are dependent on the value of the refractive index assumed for the aerosol. There are however several means to determine the size distribution of the aerosol, hence it is natural to see if given the size distribution, one could determine the refractive index. Because it is a directed search method, it has the advantage of computational efficiency, and as it is not an analytical inversion, scattering solutions based on non-spherical shapes can be readily adapted. Hodgson (2000) has shown that the GA method is very effective in retrieving the correct solution from the many possible minima in the highly convoluted solution space of a similar problem. There are other possible optimization or search methods, i.e., lookup table methods or simulated annealing, but we chose the GA as we have experience with the GA software library used in this study and because our goal is not an examination of the methods, but in the result of the retrieval.

The light intensities scattered from an ensemble of particles with incident light polarized parallel to a measurement scattering plane is very different from that with incident light polarized perpendicular to this plane. Thus, the addition of the measurement of another polarization plane contributes greatly to our understanding and verification of the principles on which our scattering algorithms are developed. As shown later, it is relatively easy to use a half wave plate to rotate the polarization plane of the incident light. With incident light polarized in one direction, it has been shown that both particle size distribution and real refractive index can be determined, although the imaginary index was less determined (Jones 1994). Although not the subject of this article, the additional polarization plane may help in determining the imaginary component. Finally, as we expect to expand this program into the examination of aerosol particles that are not homogenous and spherical, the additional scattering information is necessary for our analysis.

In this article we describe a dual polarization 21 channel polar nephelometer specifically designed for aerosol studies, followed by a description of the GA method, and an error analysis. Several GA retrieval tests using polystyrene latex (PSL) spheres and drops from ammonium sulfate in water that have well defined optical and size properties illustrate the effectiveness and limitations of the method and finally we conclude with the measurement of the refractive indices of mostly spherical organic aerosols generated in a large photochemical reaction chamber.

NEPHELOMETER DESIGN

There are 21 silicon photo diode detectors arrayed in a plane about a point where a stream of aerosol particles intersect a laser beam as shown in Figure 1. The detectors view the scattering center through 4.8 mm holes that are 50 mm long and the detectors are about 100 mm from the scattering center. The holes define an aperture, which limits the acceptance angle to 4.46° and the detectors are placed at angular locations such that each detector cannot view the sensing area of any other detector, i.e., no detector is positioned directly across the scattering plane from another detector. Also, all surfaces beyond the scattering center and within each detector's field of view are canted and polished so that light reflected from the scattering center are directed away from the detector plane, thus secondary reflections are reduced. However, significant background light levels are detected in the far forward and reverse detectors due to light leakage out of the laser beam dump and from diffraction caused by dust or imperfections in the laser beam optics that could not be completely corrected. These stray light levels change during operation due to the inherent directional beam drift, thus the stray light levels are determined periodically (about every 5 minutes) and subtracted from the signals measured when particles are present. The amplified output of each detector is directed to a data acquisition card and an embedded PC built into the instrument. About 20 times each second, about 500 analog-to-digital conversions for each channel are averaged and sent to an external PC for storage. Laser light at 670 nm from a 350 milliwatt laser diode is polarized parallel to the scattering plane, and directed across the scattering plane and into the beam dump. The polarization plane is rotated 90° every 3 seconds by a solenoid-operated half-wave plate. The degree of linear polarization is measured to be 0.96 and the laser polarization axis is aligned to be within +/–2° with respect to the scattering plane and is collimated to be about 5 mm high and about 0.5 mm wide at the intersection with the sample stream. Particles confined via a sample guide tube with a rectangular cross section are directed into the scattering plane in a swath about 19 mm long and 4 mm wide, parallel and coincident with the laser beam. Thus, the laser light interacts with the particle beam along this 19 mm long swath, and because the detector apertures define finite sensing angles, the relative intensity (RI(θ _i )) of light scattered into each channel i is determined via

where k _i is the channel-specific response function, a _d is the position a scattered photon falls onto the detector with radius of r _a , a _l is the position along the laser beam that the scattering occurs between −D _L to D _L , which are dimensional limits in which the detector can see the laser beam based on either the field of view of each detector, or for the far forward or reverse channels, the physical length of the sample in the laser beam. D(θ′, a _d , a _l ) accounts for the circular shape of the detector aperture as the amount of light getting to the detector is dependent on the height of the hole where the photon enters the detector and the scattering angle (θ′) of each photon is adjusted to account for the difference between the center of the scattering plane and the particle position along the laser beam and its position on the detector via

Theoretical angular scattering intensities, _p (θ ′), are determined for the distribution of particle sizes and for incident light polarized parallel or perpendicular to the measurement plane with corrections due to the degree of linear polarization of the laser beam. The intensity differences caused by the detector to particle distances are adjusted by the second term in the integral where R(θ _i ) is an arbitrarily chosen detector distance used to reference all other distances , and the distance from the scattering point to the detector is given by

FIG. 1 (a) Top view schematic of the optical configuration of the polar nephelometer that shows the placement of the detectors and their apertures. (b) The particles are confined by a sheath flow to the center of the detection array. An electromechanical device rotates the 1/2 wave plate 45° periodically to change the polarization plane of the incident light from parallel to the scattering plane to perpendicular to it.

These parameters are illustrated in Figure 2a. Note that an absolute calibration is not needed for the inversion method, hence only the relative intensity RI(θ _i ) is determined at each channel. At the bottom of Figure 2 is shown the instrument response for spherical PSL particles with an average diameter of 806 nm and a standard deviation of 11.28 nm. The dramatic increase in the detector response in the far forward and reverse directions is because the detector sees only a small fraction of the interaction area along the laser beam at the side angles. This adjustment assumes that the laser beam has no width, which is reasonable as the laser beam width is about 0.5 mm. This derivation also assumes that the laser beam has no height, and a similar calculation that includes the laser beam height does not affect the adjustment significantly. Multiple scattering is ignored as optical depths in the scattering volume are very small (i.e., less than 10^{− 5}) at the particle concentrations that produce reasonable detectable signals in this instrument.

FIG. 2 (a) Geometric view from above the scattering plane that illustrates the parameters of Equation (1), in which only two detector locations are shown for clarity and each detectors field of view (FOV) is shown as a dotted line. Particles are confined by the sample port to intersect the laser beam between –D _L to D _L . The detector field of view is larger than the length of the laser beam in which particles are present as the detector scattering angle approaches the forward and reverse angles. (b) Diamonds (⧫) are the theoretical expectations for PSL particles with a diameter of 806 nm (line) adjusted to instrument sensing geometry described by Equation (1) with incident light polarized parallel to the measurement plane.

Shown in Figure 3 is the average of about 15 measurements of the relative light intensities measured from PSL spheres with specified size distributions verified via counts of TEM images and that have a manufacturer specified refractive index of 1.5854–0.0i at 670 nm. These particles are generated from a solution of PSL particles in deionized and filtered water through a spray aerosolizer and then dried by passing the aerosol stream through a dessicant canister. Scanning mobility particle sizer (SMPS) measurements (TSI Model 3080) showed that the majority of particles had sizes similar to those of the calibration values, however, there was also a small concentration of very small particles (less than ∼ 0.45 μ m), which was also seen by Jones et al. (1994) in a similar experiment. Because of the small size and low concentration of these particles, the instrument can not detect them. Because the width of narrow size distributions are difficult to measure with the SMPS, TEM images of these particles have been used to verify the monodispersity of the distributions. These scattering measurements are compared to the expectations developed using Mie theory and adjusted for the geometric parameters as discussed in Equation (1). Because there is no attempt or need to determine the absolute scattered intensities for each particle, the expectations are fitted to the experimental result via a multiplicative constant developed using the method of least squares. From these measurements, detector response values (k _i in Equation [1]) are determined from the response of the instrument to the PSL particles with an average diameter of 806 nm and parallel incident light and applied to the other measurements in Figure 3. The error bars are based on the standard deviation of the 15 measurements made at each channel, which is caused by electronic noise in the detector/amplifier circuits. The differences between the expectations and the measured result at the lower signal levels are probably because of incomplete absorption of light scattered into other directions within the scattering volume. Although every effort has been made to absorb this light via specially designed beam dumps placed all around the scattering volume, the light scattered into the forward (and sometimes reverse) directions have intensities that are several orders of magnitude greater than those scattered to the sides. Note that the larger PSL spheres have a stronger overall scattering signal and smaller differences between the expectation and measurement at the lower signal levels. The large difference in the reverse direction of the 596 nm parallel measurement is because the laser beam directional drift caused a higher than normal backscattering signature. Despite these differences, the measurement clearly shows the predicted dips in the scattered intensities and follows the geometric adjustments described above by Equation (1).

FIG. 3 (a) The angular scattering intensities of PSL spheres with a specified diameter of 360 nm and standard deviation of 5.40 nm measured by the polar nephelometer with incident light polarized parallel (▴) and perpendicular (◊) to the measurement plane with expectations developed via GA search of Mie-Lorenz theory calculations and fitted to the measurement. (b) Similar results but for PSL particles with a specified diameter of 596 nm and a standard deviation of 7.75 nm. (c) Similar results but for PSL particles with a specified diameter of 806 nm and a standard deviation of 11.28 nm.

GA DETERMINATION OF REFRACTIVE INDEX

Method

Genetic algorithms (Goldberg 1989) have been used to solve intractable multivariate problems which often cannot be determined with other more conventional means. Briefly, the solution is found by searching a solution space, in this case the space of possible real and imaginary refractive indices (m_r, m _i ). A fitness test is applied to a small subset or “population” of the solution space, which is a randomly selected set of refractive index pairs. Those solutions of the population that are “better” are combined to produce “offspring” with parameters that should have a better solution to the problem. This algorithm is repeated until a preset number of generations occur or a desired level of fitness is achieved. Thus the theoretically derived scattering intensities, P _l,thy (θ _j ) determined at each discrete angle (θ _j ) for particles with a refractive index of (m_r, m_i), is compared to the experimental results, P _l,meas (θ _j ), via a fitness test modeled after that used by Lienert et al. (2003) as follows:

for each scattering channel from 1 to N, l indicates the polarization state of the incident light and the log factor ensures that the fitness is not biased towards higher intensities. A final fitness value is determined via F = F _⊥ + F _II, with ⊥ for perpendicular incident light and II for parallel incident light. Theoretically determined angular scattering properties are adjusted to match the experimental setup described in Equation (1) and fitted to the measurement using a least squares fitting method. In order to fully explore the GA capabilities, the search space is made large, i.e., the real space ranges from 1 to 2, while the imaginary space ranges from 0 to 10^{− 5}. Faster convergence to the solution is expected if limits to the search space, i.e., the measured particle are known to absorb, can be applied. It is assumed that the particle size distribution is predetermined via another instrument, such as a scanning mobility particle sizer (SMPS). Both parts of the refractive index are represented in the GA routine by mapping each to a binary representation that can be adjusted in size (i.e., from 8 bit and up) to any desired resolution. Single-point crossover is used to combine the results of two binary representations in which the bits of two parents are swapped about a randomly selected point in the binary representation. A mutation factor of 0.9 periodically alters bits in the binary representation so that the search space is not confined to a local minimum and elitism ensures that the best result is kept at each generation. After each generation, duplicate population members are eliminated by replacement with a randomly generated member.

The software for this work used the Galib genetic algorithm package, written by Wall (1996) at the Massachusetts Institute of Technology. This search algorithm is a directed search method and not an analytical solution hence it does not require an inversion of the scattering solution. Thus other algorithms that determine the scattering properties of non-spherical particles, i.e., finite difference time domain (Yang et al. 2000) or discrete dipole (Fuller 1994) could be used.

Error Analysis

In this section we discuss some possible error sources and how they affect the retrieval. The retrieval algorithm does not require an absolute calibration of the retrieved intensities hence eliminating a large source of systematic errors. Thus we examine errors arising from the particle sizing uncertainties and electronic noise which induces random variations of the measured intensities. These tests are performed on only the retrieval of the real refractive index, as the retrieval of the imaginary index requires longer computer processing times. Synthetic experimental results are produced using Mie-Lorenz theory.

Shown in Figure 4 are the retrieved real refractive indices from synthetic experimental data with noise applied to each channel via

where RI(θ _ι) is the original non-noisy synthetic signal at scattering angle i, RI′(θ _i ) is the noisy result, N _max is the noise level and R is a random number between 0 and 1. N _max is based on the signal to noise ratio (SNR) of measurements of PSL spheres, which varies from about 1.3 to over 350 depending on the sample particle concentration, particle size, and the scattering angle. As shown in Figure 4, the error in the retrieved m_r is less than 0.04 for N _maxbased on a SNR of 1.3, because the GA routine does an effective job of determining the solution that best fits the available data despite the noise.

FIG. 4 GA retrieved real refractive index for synthetic experimental results based on PSL spheres with an average particle radius of 180 nm and a standard deviation of 0.054. A random noise level based on the maximum SNR is applied to each channel to produce the error bars that are the standard deviation of 10 separate GA retrievals of the simulated noisy results. The dotted line is the correct m_r, which is 1.5854.

The effect of errors in the size distribution measurement are shown in Figure 5a, in which the GA retrieved real refractive index is plotted as a function of percentage deviation in the distribution mean determined via D′ _pi = D _pi + Δ/100D _pi , where Δ is the percentage level of change and D′ _pi is the adjusted value. PSL spheres with a mean diameter of 360 nm and three different normal distribution standard deviations of 3.6, 36.0, and 72.0 nm is assumed. These correspond to %coefficient of variation values (%CV = 100 × standard deviation/mean) of 1%, 10%, and 20%, which provides a measure of the width of the particle size distribution. The narrower distributions are usually encountered only in laboratory conditions. Each point in the plot is the average of 10 separate GA searches each of which differed very little indicating that the GA routine has converged to a solution. With no error in the size distribution measurement, m_r = 1.5837 ± 0.0027, which is very close to the expected 1.5854. As shown, the error in the retrieved real refractive index is smaller for wider distributions as the scattered intensities are more dependent on the particle refractive index and less on the mean particle size, i.e., the scattering properties of a completely flat distribution would be dependent only on the refractive index. In real situations the distribution widths are much wider and often multimodal and with corresponding lower errors in the m_r retrieval as shown in Figure 5b in which a similar analysis is done on a representative urban air three-mode distribution described by Jaenicke (1993). For comparison, a similar analysis for a narrow mono-modal lognormal distribution with a mean diameter of 0.1 μ m and a standard deviation of 0.05 is also plotted. It is assumed that the particles are spherical and have a refractive index of 1.5–i0.0. There is very little variability in the retrieved three mode distribution m_r (±0.002) even though the errors in the mean size are up to 20%. Part c of Figure 5 shows this same analysis but with errors simulated in the width or standard deviation of the size distribution. Errors in the GA retrieved refractive index are also minimal for the wider urban distribution when simulated errors in the distribution variance range up to 20%.

FIG. 5 (a) The change in the GA retrieved m_r as a function of the percent error in the measured mean of the particle size distribution. Three different standard deviations are used corresponding to %CV of 1%, 10%, and 30%. These results are based on the retrieval of synthetic experimental results determined using Mie-Lorenz calculations based on PSL spheres. (b) Same as above but for the multi-modal urban air distribution described by Jaenicke (1993), and a mono-modal lognormal distribution of spheres with a mean diameter of 0.1 μ m, σ = 0.05 μ m and m_r = 1.5. (c) Same as b, but for the GA retrieval of m_r as a function of errors in the variance of the distributions.

EXPERIMENTAL RESULTS

PSL Spheres

In Table 1 are shown the GA retrieved real refractive indices from PSL spheres based on polar nephelometer measurements as discussed above and illustrated in Figure 3, along with the manufacturer's model numbers, specified size distributions and SMPS measured mean diameters. The smallest measured SNR was about 5.3 for the smallest particles (diameter = 360 nm) which had the lowest overall signal. As these particles have negligible absorption, the GA search was limited to the m_r, with a population of 20, a mutation factor of 0.9 and ran for 10 generations. The GA search used the size distribution specified by the manufacturer, and it is noted that SMPS measurements of the particle mean diameters are within 5% of these values. The size distributions for these particles is very narrow, (%CV ∼ 1.5% in each of these cases), hence as discussed above errors due to uncertainties in the distribution mean dominate, which is about ±0.1, based on a 5% error in the distribution mean. The error of the retrieved m_r for the particles with a diameter of 596 nm (m_r = 1.65) that differed the most from the expectation is within that predicted by the error analysis for the very narrow distribution of Figure 5a. As noted above and in Figure 3b, this measurement deviated more from the expectation than the measurement for the 806 nm PSL spheres. There is also a significant difference between the expectation and the measurement for the 360 nm and 596 nm diameter spheres above about 120° for both the parallel and perpendicular component, which is most likely due to multiple scattering within the chamber as previously discussed. However, the GA routine was able to retrieve reasonable m_r values as it was constrained by the specified particle diameter, i.e., it determined the result that produced the lowest fitness value using the majority of correct angular scattering measurements at the diameter of 360 nm (or 596 nm).

TABLE 1 GA retrieved real refractive indices from polar nephelometer measurements of PSL spheres

Part number*	Manufacturer specified average diameter	Manufacturer specified standard deviation (nm)	SMPS Measured particle diameter (nm)	GA retrieved mr
5036A	360 nm	5.40	359	1.56 ± 0.1
5060A	596 nm	7.75	594	1.65 ± 0.1
5081A	806 nm	11.28	763	1.59 ± 0.1

Ammonium Sulfate

Ammonium sulfate (NH₄)₂SO₄-H₂0 forms spherical drops and the refractive index is dependent of the amount of solute dissolved in the water drop and the amount of water in the drop is a known function of the ambient relative humidity (Tang and Munkelwitz 1991; Tang 1996). An aerosol of these particles is produced as described above for the PSL particles, but with a solution of 0.125% by weight ammonium sulfate and water in the spray aerosolizer and the relative humidity is changed by varying the amount of silica gel in the drier. The aerosol stream is first monitored for the relative humidity value then the aerosol output is connected to the polar nephelometer, then the SMPS, and finally the relative humidity is measured again. As the silica gel effectiveness changes during the experiment, the relative humidity increases by about 2–7% during the 9 to 10 minutes required for these measurements. Shown in Figure 6 are the scattered intensities measured by the polar nephelometer when the relative humidity changed from 59% to 61%, (part a) and from 40% to 47% (part b) during the course of the experiment along with theoretical expectations based on Mie-Lorenz calculations derived from a GA search. Error bars are based on the standard deviation of 10 separate intensity measurements. During the three SMPS measurements taken in succession the measured mean of the particle radii increased about 6 to 7 nm due to the increase in humidity. Thus, the GA routine was set up to search for both the real refractive index and the particle size mean and standard deviation. Limits on the size distribution search space are determined by the measured mean radius of 55 to 65 nm for the case in which the RH is about 60% and 50 to 60 nm for the case in which the RH was about 40%. Similar limits on the standard deviation search space are 47 to 53 nm for the RH ≅ 60% case and 44–48 nm for RH ≅ 43%. In each case the GA population was 100 and it ran for 10 generations. The average retrieved refractive index and the range of 10 separate searches are summarized in Table 2. The retrieved particle radii were near the upper limit of the search range, which is reasonable given that the relative humidity and thus particle size was higher as the nephelometer measurements are taken before the SMPS measurements. The expected value for m_r is derived from the parameterization, m_r = 1.333 + 1.673 × 10^{− 3} x −3.95 × 10^{− 6} x ² (Tang and Munkelwiotz 1991) where x is the solute weight percent that is determined from the plot for particle mass change versus relative humidity in Tang (1996). In both cases the retrieved refractive index is close to the expected value with fitness values of about 0.94, where it is noted that a perfect match of theory to experiment would return a fitness value of 1.0. On the intensity scattering plots of Figure 6 most of the measured points fall near the expectation, except in the reverse direction above 120°, the measurement tends to increase above the expectation. As with the PSL spheres, the GA search was able to find the correct result despite this bias because it was limited by the size distribution parameters.

FIG. 6 (a) Nephelometer measured scattering properties for incident light polarized parallel (▴) and perpendicular (◊) to the scattering plane for (NH₄)₂SO₄-H₂O droplets when the relative humidity was about 60%. Error bars are based on the standard deviation of 10 separate measurements. The lines are the Mie-Lorenz scattering properties for spheres with incident light polarized parallel and perpendicular to the scattering plane and the GA determined real refractive index. (b) Same as b, but when the relative humidity was between 40 to 47%.

TABLE 2 GA retrieved real refractive index from the experimental scattering results measured from (NH₄)₂SO₄-H₂O droplets seen in Figure 6

% Relative humidity	Expected mr	GA retrieved mr	GA fitness
59–61%	1.414–1.413	1.414 ± 0.03	0.945 ± 0.01
40–47%	1.434–1.427	1.425 ± 0.02	0.944 ± 0.001

α-Pinene

We chose to study the secondary organic aerosol (SOA) particles formed by α -pinene as it is considered to be the most abundant SOA forming compound (Geron et al. 2000) and has been studied extensively (i.e., see the references of Saathoff et al. 2003). To generate secondary organic aerosol (SOA), 0.60 ppm α -Pinene and 0.54 ppm NO_x in a 24 m³ Teflon chamber was exposed to solar radiation. After about an hour, SOA homogeneously nucleated and quickly grew to several hundred nanometers. The nephelometer measured the scattering properties of the SOA and the results are shown in Figure 7 along with a representative transmission electron microscope image. The concurrently measured SMPS size distribution has an average diameter of about 457 nm with a standard deviation of about 233, thus the size distribution %CV, or width, is about 50%. This predicts a minimal error in the m_r retrieval on errors in the SMPS measurements as shown in Figure 5a. The images were not taken concurrently with the nephelometer measurements, hence the size differential, however these, and other similar images, do illustrate that the SOA particles are spherical, as expected. The nephelometer results are the average of 20 measurements taken over a period of about 5 minutes, during which the growing particle's average diameter changed slightly (+0.4 nm) and the smallest SNR value based on the standard deviation of these measurements was about 4.3. Using a population of 20, and mutation rate of 0.9, after 10 generations, the GA retrieved real refractive index, m_r is 1.42 ± 0.02 in which the error is based on the smallest SNR derived from the standard deviation of the 20 nephelometer measurements and the error analysis detailed for Figure 4. However, as seen in Figure 7, the GA determined scattering properties do not match the measured results perfectly, particularly for the perpendicular measurement between 150° and 170°. This discrepancy may arise from non-homogeneity in the particles, which will cause a deviation from Mie-Lorenz theory. Since the photochemistry that generated the particles is dynamic, it is reasonable to assume that the material that forms the core of the particles is different from the material that causes their growth, although little is known about how the material nucleating or condensing changes as the particles grow and the oxidation chemistry in the chamber evolves.

FIG. 7 (a) Nephelometer measured scattering properties for incident light polarized parallel (▴) and perpendicular (◊) to the scattering plane of SOA particles generated from the photochemical reaction of α -pinene. The lines are the GA determined scattering properties for homogenous spheres with m_r = 1.43 fitted to the experimental results using the least squares method. (b) Representative TEM image of the SOA particles.

The only existing measurement of an α -pinene SOA refractive index was derived for particles generated by oxidizing α -pinene with O₃ in a chamber (Schnaiter et al. 2005), which differs somewhat from our experiments, which oxidized α -pinene with OH, O₃ and NO₃ in the presence of NO. Schnaiter et al. (2005), however, found a refractive index, 1.5, similar to what we derive using the Mie-Lorenz model to match multi-spectral extinction, hemispheric backscattering and Ångstrøm exponent measurements.

SUMMARY

It has been shown from experimental measurements that the dual polarization polar nephelometer and the GA search method described in this manuscript can derive m_r of homogenous spherical aerosol particles when simultaneous size distribution measurements are included. An error analysis based on synthetic experimental results has shown that the GA method can determine the m_r even with a minimum SNR as low as 1.3, and if the particle size distribution is sufficiently wide, errors in the m_r retrieval are small. GA retrieval of the real refractive index of various mono-dispersions of polystyrene latex spheres and ammonium sulfate diluted in water drops show that this method is effective. However, particles in the real world rarely follow this model, hence aerosol m_r retrievals based on the Mie-Lorenz theory are suspect, as shown by an analysis of SOA produced by α -pinene. Initial results (not shown) indicate that the imaginary component of the refractive index can also be retrieved with this GA method, hence an error analysis is being conducted.

As the GA retrieval method described here is not an analytical inversion, algorithms that model the scattering properties of particles that are not homogenous spheres can and will be adapted in order to determine the refractive index of these particles as most of the aerosols in nature are either non-spherical or inhomogenous spheres.

Acknowledgments

We would like to thank Yoji Reichert for his help with the SOA experiments and the two anonymous reviewers for their very helpful suggestions. This research was supported primarily by the U.S. Department of Energy's Atmospheric Science Program (Office of Science, BER, Grant No. DE-FG02-05ER64011).

Notes

*Duke Scientific Corp.