April 1982 A. Mita 765. Light Absorption Properties of Inhomogeneous Spherical. By Akiyoshi Mita*

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1 April 1982 A. Mita 765 Light Absorption Properties of Inhomogeneous Spherical Aerosol Particles By Akiyoshi Mita* Water Research Institute, Nagoya University, Nagoya 464, Japan (Manuscript received 12 September 1980, in revised form 28 September 1981) Abstract Light absorption properties of inhomogeneous spherical particles, each consisting of absorbing core and nonabsorbing shell, were investigated using the theory of light scattering from a coated sphere. The absorbing core was assumed to be made of either carbon soot or hematite (Fe2O3), both of which are considered to be important substances in determining the absorptivity in the visible region of atmospheric aerosols. The absorption properties of polydisperse systems in which these inhomogeneous particles are distributed by the power law size distribution were also investigated. The absorption cross section of spherical soot and hematite particles can increase by a factor of about two when they become coated with concentric spherical shells of nonabsorbing substances having real refractive indices This enhancement of absorption occurs for soot particles of any size. For hematite particles, however, in some cases the nonabsorbing coating acts to reduce the absorption cross section in the region where the particle size is comparable to the wavelength of light. It is shown that the imaginary part of the effective refractive index for polydisperse systems composed of absorbing and nonabsorbing substances becomes somewhat larger when the absorbing component occurs as mixed particles coated with nonabsorbing substances. 1. Introduction In recent years there has been a growing concern about the light absorptive properties of atmospheric aerosols. Observational data suggest that the amount of solar radiation absorbed by particulate matter is of comparable magnitude with that absorbed by gaseous constituents, such as water vapor (Kondratyev et al., 1974; Major, 1976). The importance of the absorptivity of aerosol. particles in determining the reflectivity of the earth-atmosphere system has been shown by several authors (e.g., Yamamoto and Tanaka, 1972). Atmospheric aerosols are in general composed of particles of diverse substances. For visible radiation, such highly opaque substances as soot and hematite (Fe2O3) may play an essential role in determining the absorptivity of the aerosols (Volz, 1972; Kondratyev et al., 1974). Recently Present affiliation: Kobe Marine Observatory, Nakayamate, Chuo-ku, Kobe 650, Japan Rosen et al. (1978) demonstrated using Raman spectroscopy technique that the optically absorbing component in urban aerosols is graphitic soot. On the assumption that these absorbing substances and other nonabsorbing substances exist as different particles in aerosols (i.e., the assumption of external mixing), it is easy to calculate the optical properties of the aerosols by applying Mie theory to given size distribution of spherical particles (e.g., Mita and Isono, 1980). In the real atmosphere, however, such microphysical processes as coagulation, adsorption of reactive gases onto the particle surface, and incorporation in the cloud forming process eventually lead to a mixed nature of individual aerosol particles (i.e., internal mixing becomes predominant). From the analysis of the CAENEX data, Kondratyev et al. (1979) suggested a possibility that hematite was embedded within other less absorbing particles in the atmosphere. The absorption property of an inhomogeneous particle which contains both absorbing and nonabsorbing

2 766 Journal of the Meteorological Society of Japan Val. 60, No. 2 substances may be different from that of a homogeneous particle composed of the absorbing substance only; therefore it is of interest to inquire into the optical properties of these inhomogeneous aerosol particles. The theory of scattering of electromagnetic waves from an inhomogeneous stratified sphere was worked out by Aden and Kerker (1951). Based on this theory extensive calculations have so far been made concerning the problem of microwave scattering from melting hailstones. Unfortunately, only very limited applications have been made to the problem of light scattering by cloud and aerosol particles. Fenn and Oser (1965) calculated the efficiency factors and angular scattering characteristics of concentric soot-water spheres in an attempt to understand the effect of soot particles incorporated in cloud droplets. The scattering and absorption characteristics of water-coated particles, which are to be encountered in high air humidity conditions, were studied by Prishivalko and Astafyeva (1974). Clearly, more detailed study is needed on the optical properties of inhomogeneous particles in order to improve our knowledge of the absorption property of atmospheric aerosols. It is the purpose of this paper to examine how the absorption properties of soot and hematite particles for visible radiation are altered when they become coated with nonabsorbing substances such as liquid water or ammonium sulfate. These inhomogeneous particles (hereafter also referred to as mixed particles) are modeled as having a two-layer structure with spherical symmetry. In this paper, details of the absorption cross section of individual inhomogeneous spherical particles with soot and hematite cores are presented for various particle sizes. Some features associated with very small and large particles are described and interpreted in terms of the Rayleigh and ray optics approximations. Next, we examine the absorption properties of polydisperse systems in which these mixed particles are distributed by the power law size distribution. The imaginary part of the effective refractive index for these internally mixed aerosols is calculated and compared with that for the externally mixed aerosol having the same chemical composition. 2. Method of calculation Fig. 1 shows the model of a mixed particle used in this study. The amplitude coefficients Fig. 1 Coated sphere model used in this study. m1 and m2 are the complex refractive indices of the core and the shell, respectively. an and bn of the electromagnetic wave scattered from such a coated sphere may be expressed as (Kerker, 1969) where n is a positive integer and the following operators are used: Other necessary operators can be defined similarly by substituting for the x and c in the square bracket and correspondingly in the expressions on the right hand side of the equations. In the expression (2), m1 and m2 are the complex refractive indices of the core and the shell relative to the external medium, respectively; a and v are the size parameters defined as where a is the core radius, b is the outside radius of the shell, and * is the wavelength of the incident light; (*, x*, and *n are Ricatti-Bessel functions with the relation *n = *n + ixn. The primes on the right hand side of (2) indicate differentiation with respect to the argument of

3 April 1982 A. Mita 767 Fig. 2 Absorption cross section of mixed particles consisting of the soot core and nonabsorbing shell for various core sizes ((a)-(e)). The core sizes are fixed, and the absorption cross section relative to the value for the uncoated core (*/*0) is plotted as a function of the shell-to-core radius ratio b/a. Solid, dashed-dotted, and dashed lines correspond to different refractive indices of the nonabsorbing shell. the function. Using the coefficients an and bn given by (1), one can calculate the extinction and scattering cross sections and other light scattering quantities as in usual Mie theory. Checks of our calculation scheme were made by setting some of the parameters a, *, mi, and m2 so that the coated sphere reduces to a homogeneous sphere and comparing the results with those obtained by the calculation scheme for a homogeneous sphere. All calculations were performed on. the digital computer using double precision arithmetic. The accuracy was checked against numerical values reported in the literature (Kerker et al.,.1962; Espenscheid et al., 1965). In the present calculation, the absorbing core is assumed to be made of either carbon soot (m1= i) or hematite (mi = i). * As for the nonabsorbing shell, three real refrac- * These values of the optical constants are for a wavelength of 0.55 j m (Dalzell and Sarofim, 1969; tive index values of 1.33, 1.43, and 1.53 were considered to cover the range of refractive indices of nonabsorbing substances in atmospheric aerosols. These values correspond to the refractive indices in the visible region of liquid water, sulfuric acid, and ammonium sulfate, respectively. In view of the nonsphericity of actual aerosol particles, the model used here will be an idealization; however, it allows a rigorous solution from which we can gain a useful insight into the optical properties of inhomogeneous particles. 3. Absorption properties of single particles 3.1 Absorption cross section Calculations were performed for mixed particles having cores of various sizes with varying shell thickness. Some of the results of these Ivlev and Popova, 1973). Carbon soot has fairly constant values in the visible region, while hematite has larger imaginary refractive index as the wavelength approaches the ultraviolet.

4 768 Journal of the Meteorological Society of Japan Vol. 60, No. 2 Fig. 3 As in Fig. 2 except for mixed particles consisting of the hematite core and nonabsorbing shell. calculations are presented in graphical form to illustrate the principal optical behavior of such particles. Figs. 2a-2e show the absorption cross section of mixed particles having soot cores of five different sizes. In these figures the core sizes are fixed, and the absorption cross section relative to the value for the uncoated core is shown as a function of the shell-to-core radius ratio, b/ a. It is apparent that the nonabsorbing coating has an effect of increasing the absorption cross section for soot cores of all sizes. When the soot core is sufficiently small (a=0.2), the absorption cross section increases monotonically with increasing shell thickness. As the core becomes large, the curves of the absorption cross section begin to oscillate. For large cores (a*4), the absorption cross section first increases to a certain value and remains fairly constant with further increase of the shell thickness. This saturated value depends on the refractive index of the shell: the larger the refractive index, the larger the increase in the absorption cross section. The absorption cross section of mixed particles with hematite cores is shown in Figs. 3a-3e. As in the case of mixed particles with the soot core, the absorption cross section increases monotonically with increasing shell thickness when the hematite core is sufficiently small (Fig. 3a). For particles whose sizes are comparable to the wavelength of light, the absorption curves become quite complicated. Note that in some cases the nonabsorbing coating has an effect of reducing the absorption cross section (*=1.9 and = 4.1). The curves for a large hematite core * (*= 10) are similar to the corresponding curves for mixed particles with a soot core, though the oscillations of the curves are more pronounced. It should be pointed out here that for wavelengths near the ultraviolet the absorption features may be a little different from those in Fig. 3, since the imaginary part of the complex refractive index of hematite takes a value larger than that used in this study. In Figs. 2 and 3 the relative absorption cross

5 April 1982 A. Mita 769 sections are presented for b/a up to 5. The absorption cross sections for larger values of b/ a will be of some interest, since in the case of cloud droplets which contain insoluble aerosol particles the ratio b/ a may reach several tens. Our calculation for the water coating (n=1.33) was performed for b/a up to 50. The result of the calculation, though not presented in this paper, shows that the behavior of the curves seen in Figs. 2 and 3 for moderate to large core sizes does not change as the ratio b/a gets far largerie., oscillations around a certain value continue up to at least b/a=50. Fig. 5 As in Fig. 4 except for hematitewater particles. 3.2 Absorptivity (1-*0) While the extinction cross section of a mixed particle increases without limit with increasing particle size, its absorption cross section remains fairly constant beyond a certain value of the shell-to-core radius ratio b/ a. Hence the absorptivity 1-*0*(*0 is the albedo for single scattering) of a mixed particle is expected to decrease with increasing shell thickness in the region where b/a is large. Fig. 4 shows the absorptivity of soot-water particles of fixed core sizes as a function of b/a. The curves for another shells with refractive indices 1.43 and 1.53 are not shown, but are similar to those in Fig. 4. It can be seen that the absorptivity decreases monotonically with increasing shell thickness for particles of small and moderate core sizes (a = 0.2, 1.9). The absorptivity of a large particle (a =10) on the whole also decreases. A remarkable feature in this case, however, is the increase of absorptivity for b/a<1.2. Fig. 5 shows the absorptivity of hematite-water particles. Features are similar to those in Fig. 4 except for the lower absorptivity of these particles. It is noted that in the curve for a =10 there exists a region where the absorptivity of a coated particle is higher than that of the uncoated core. A small dip near b/a=1 corresponds to the relatively slow increase of the absorption cross section in this region of b/a (compare Fig. 2e and Fig. 3e). 3.3 Interpretation of the results for very small and large particles We have noticed that both soot and hematite particles have a fairly simple feature of absorption cross section in common when the particles are either very small or large. For very small particles, an analytical expression for the absorption cross section can be obtained (the Rayleigh approximation), while geometrical optics is applicable to very large particles. It will be useful to consider these two limiting cases in order to have a better understanding of the results obtained in section 3.1. When the size of an absorbing sphere coated with a concentric spherical shell of nonabsorbing substance is sufficiently small compared to the wavelength of light, its absorption cross section may be expressed by the following formula (see Appendix A): Fig. 4 1-*0(*0 is the single-scattering albedo) as a function of the shell-to-core radius ratio for soot-water particles of three different core sizes. 1-*0 equals the ratio of the absorption cross section to the extinction cross section. where hubs is the absorption cross section, q(= a/b) is the ratio of the core radius to the radius of the total particle, and a is the size parameter of the core as defined earlier. A, B, C, and D are constants dependent on the complex refractive indices of the core and the shell:

6 770 Journal of the Meteorological Society of Japan Vol. 60, No. 2 where n1 and k are the real and imaginary parts of the refractive index of the absorbing core, and n2 is the refractive index of the nonabsorbing shell. A, B, and D are always positive while C can be either positive or negative depending on the values of n1, n2, and k. The general behavior of the expression (4) as a function of q is complex; however, it becomes simple when n1>n2, the case we are treating. In this case the constant C also becomes positive and we can easily see that *abs increases monotonically to a certain value as q decreases from unity (i.e., b/a increases from unity) for a fixed value of a. Using the refractive index values of soot and hematite, we can predict on the basis of the expression (4) that the absorption cross section of soot and hematite particles of very small size increases by a factor of 1.13 and 1.52 (asymptotic values as q*0), respectively, when they become coated with water. This explains the behavior of the curves for a=0.2 in the region where the shell thickness is small (Figs. 2a and 3a). Further increase in the absorption cross section with increasing shell thickness as seen in these figures cannot be explained by the present approximation, since the size of the total particle becomes too large for the Rayleigh approximation to be applicable. On the other hand, there can be some cases when *abs decreases with increasing shell thickness provided that nl<n2. For example, on the basis of the expression (4) the decrease in absorption by a factor of 0.76 is predicted for a very small soot particle (n1=1.57) when coated with a hypothetical substance having a real refractive index of n2 = 2.0. In the real atmosphere, however, such cases are rarely expected. Therefore we conclude that in most cases the absorption cross section of a very small absorbing particle more or less increases when it acquires a shell of nonabsorbing substances. Next, we will turn to interpreting the results obtained for large particles. As is well known, the laws of geometrical optics can be applied to particles sufficiently large compared to the wavelength. Fig. 6 illustrates the "lens action" of the Fig. 6 Illustration of the "lens action" of a spherical dielectric shell covering a large absorbing core. spherical dielectric shell covering a large absorbing core. The optical cross section of an absorbing core of fixed size gradually increases as the thickness of the shell increases from zero, since all the light rays incident on the shell are refracted inward to fall onto the inner core. When the shell attains to a certain thickness, the rays incident on the rim of the shell become tangent to the inner sphere after refraction. Shown in Fig. 6 is this critical state. Further increase in shell thickness will not act to enhance the absorption because the rays incident near the rim of the shell are refracted to pass by the inner core without falling on it. The critical ratio of the radius of the shell to that of the core at which the absorption ceases to increase can be where n2 is the refractive index of the shell. Hence it follows that the larger enhancement of absorption occurs for the larger refractive index of the shell. The above consideration based on geometrical optics well explains the absorption features observed for mixed particles having fairly large soot and hematite cores (Figs. 2e and 3e). The phenomenon that a thin water shell covering a large soot particle tends to increase the absorption was first recognized by Fenn and Oser (1965). This enhancement of absorption has been speculated to be due to a focusing of the incident radiation onto the absorbing core by the dielectric coating (Kerker, 1969; Kattawar and Hood, 1976). The consideration given in this paper confirms this mechanism of the enhance-

7 April 1982 A. Mita 771 ment of absorption. It is interesting to note that this focusing effect exists also in the case of particles whose sizes are not so large compared to the wavelength (*=4.1). 4. Absorption properties of polydisperse systems 4.1 Model size distribution o f mixed particles In the preceding section we have examined the absorption property of individual mixed particles having soot or hematite cores. It is expected that in the real atmosphere these particles occur as polydispersions. The size of soot particles produced by combustion processes ranges from the submicron to visible (Finfer, 1967), though particles with radii larger than a few microns will total number of aerosol particles. Three values of 2.5, 3.0, and 3.5 are chosen for the exponent v *. For the distribution of the thickness of the spherical shell, we have considered the following two cases. Case I: The shell volume is proportional to the surface area of the core. Case II: The shell volume is proportional the volume of the core. In case I the relative thickness of the shell as compared to the core size increases with decreasing particle size; in case II it is constant, independent of the particle size. The thickness of the shell of individual particles can be determined for each of these two cases provided that the ratio of the total volume of the shell material to that of the core material summed up over the size distribution of the mixed particles is specified. In the following part of this paper, quickly fall out. As for hematite particles, Kondratyev et al. (1979) found their sizes to be about 0.1*m; however, the existence of hematite particles in the supermicron size range is this ratio is expressed by the parameter p (see Appendix B). also reported (Klappenbach and Goranson, 1979). In spite of these reports, our knowledge of the 4.2 The amount o f light absorbed size distribution of these absorbing particles is Table 1 gives values of the absorption coefficient still very incomplete. (i.e., the sum of absorption cross sections In the present study, the same expression as of individual particles) calculated for polydisperse used in our previous work is adopted for the systems of mixed particles consisting of size distribution of soot and hematite cores: the soot core and (NH4)2SO4 shell (n2=1.53). The calculation was performed for a wavelength of 0.55*m. The total volume of soot is fixed as 30*m3/cm3. The parameter p in this table denotes the ratio of the total volume of (NH4)2SO4 where r is the radium (pm) of the core of mixed to the total volume of soot: thus, p=0 means particles, N is the cumulative number of particles that soot particles exist without the (NH4)2SO4 per unit volume of air, C1 and C2 (C1= coating. For each of p=0.5 and p=1.0, the 0.08_* x C2) are constants dependent on the absorption coefficients corresponding to the two different cases of the shell thickness distribution are shown. The values in parentheses are the absorption coefficients divided by the value for Table 1 Absorption coefficients at the wavelength of 0.55*m of polydisperse systems of mixed particles with the soot core and (NH4)2SO4 shell. Values in parentheses are the absorption coefficients divided by the value for the uncoated soot aerosol (p=0). See text for the meaning of the parameter p. to

8 772 Journal of the Meteorological Society of Japan Vol. 60, No. 2 Table 2 As in Table 1 except for polydisperse systems of mixed particles with the Fe2O3 core and (NH4)2SO4 shell. the uncoated soot aerosol. From this table it is apparent that the absorption coefficient of soot aerosol is larger when soot particles occur as mixed particles with the (NH4)2SO4 shell. This can reasonably be expected from the results obtained for single particles in section 3. It is noted that the enhancement of absorption by the (NH4)2SO4 shell becomes remarkable as the exponent ** decreases: in the case of ** = 2.5, the absorption coefficient for p=1.0 is larger by a factor of about 1.5 than the corresponding value for the uncoated aerosol. The absorption coefficients for polydisperse systems of mixed particles consisting of the hematite core and (NH4)2SO4 shell are shown in Table 2. Features similar to those in Table 1 can be observed, though in this case the enhancement of absorption by the (NH4)2SO4 coating is small. In section 3 we have seen that there are cases when the nonabsorbing coating has an effect of reducing the absorption cross section of hematite particles. When averaged over power law size distributions, the overall effect turns out to be increasing the absorption. The above results are only for mixed particles with the (NH4)2SO4 shell (n2=1.53). In view of the results obtained in section 3, however, the similar would be true also for other types of mixed particles with different shells (n2=1.33 and n2=1.43). It is reasonable to expect that in these cases the enhancement of absorption by the nonabsorbing coating will be less remarkable. 4.3 Imaginary part o f the effective complex refractive index Based on an aerosol model in which soot or hematite is externally mixed with nonabsorbing substances, Mita and Isono (1980) calculated the effective complex refractive index (i.e., average complex refractive index) of the two-component aerosol to find relationships between the imaginary part and the size distribution as well as the content of these absorbing substances. In this paper we present the results of simialr calculations performed for another aerosol model in which soot or hematite is internally mixed with nonabsorbing substances within aerosol particles. The nonabsorbing substances are represented by (NH4)2SO4. The aerosol model consists of two kinds of particles-absorbing and nonabsorbing. The absorbing ones are mixed particles considered in section 4.2. The nonabsorbing ones are pure (NH4)2SO4 particles, which are assumed to follow a power law size distribution with ** = 3.0 in the expression (7). The effective complex refractive indices which give correct values of both the extinction coefficient and the single-scattering albedo of these two-component aerosols were calculated for a wavelength of 0.55*m. The detailed procedure to obtain the effective refractive index has already been described in Mita and Isono (1980). Fig. 7 shows the imaginary part of the effective complex refractive index for soot-(nh4)2so4 aerosols plotted as a function of the soot content. Shown in this figure is the result for aerosols containing mixed particles the soot cores of which obey the power law size distribution with *=3.0. The parameter p is the measure of * the degree of internal mixing and has the following meaning: an aerosol containing 10% soot with p=0.5, for example, means that out of 90% (NH4)2SO4, 5% (=0.5*10%) exists as the outer shell of soot particles and 85% exists as pure (NH4)2SO4 particles. It can be seen from this figure that the imaginary refractive index for internally mixed aerosols is larger than that for

9 April 1982 A. Mita 773 Though the above are only for aerosols whose absorbing component (soot or hematite) obeys the size distribution with **=3.0, similar results would be presented also for other cases of ** = 2.5 and **=3.5. It can be expected that the effect of internal mixing will be relatively large for ** = 2.5 and in contrast small for ** = 3.5, as can be judged from Tables 1 and Summary and concluding remarks Fig. 7 Imaginary refractive index at the wavelength of 0.55*m of soot-(nh4)2so4 aerosols as a function of the soot content for different degrees of internal mixing (p=0, 0.5, 1.0). Dashed-dotted and dashed lines correspond to case I and case II of the distribution of the shell thickness of mixed particles, respectively. The solid line represents the imaginary refractive index for the externally mixed aerosol (uncoated aerosol). The size distribution of the soot core is the power law with *=3.0. * Fig. 8 As in Fig. 7 except for hematite- (NH4)2SO4 aerosols. The size distribution of the hematite core is the power law with v*=3.0. the corresponding externally mixed aerosol. This increase in the imaginary refractive index amounts to about 50% for p=1.0. The difference caused by two different cases of the distribution of the shell thickness is small. The imaginary part of the effective refractive index for hematite-(nh4)2so4 aerosols is shown in Fig. 8. It can be seen that the imaginary part of the refractive index for internally mixed aerosols is slightly larger than that for the externally mixed areosols. The absorption property of inhomogeneous spherical particles for visible light was investigated using the exact theory of light scattering from a two-layered particle with spherical symmetry. The results for single particles can be summarized as follows: 1) When a spherical soot particle becomes coated with a concentric spherical shell of nonabsorbing substances having real refractive indices , its absorption cross section increases by a factor of about two at the most. This enhancement of absorption occurs for a soot particle of any size. 2) In most cases the absorption cross section of a spherical hematite particle also increases when it becomes coated with a nonabsorbing shell. However, the behavior in this case is rather complex. There are cases when the nonabsorbing shell has an effect of reducing the absorption cross section for hematite particles whose sizes are comparable to the wavelength of light. 3) Both soot and hematite particles have a fairly simple feature in common when the particle size is either very small or large. In this case the enhancement of the absorption cross section by the nonabsorbing coating occurs. The mechanism of the enhancement of absorption for fairly large particles (size parameter *5) can be interpreted as the lens action of the nonabsorbing shell. 4) The absorptivity 1-*0 (*0 is the albedo for single scattering) of these particles in general decreases with increasing thickness of the nonabsorbing shell. For fairly large particles, however, there exists a range of the shell thickness where the absorptivity as well as the absorption cross section of coated particles becomes larger than that of the uncoated core due to the lens action of the spherical shell. The absorption property of polydisperse systems of these mixed particles was then studied. The overall effect of the nonabsorbing coating,

10 774 Journal of the Meteorological Society of Japan Vol. 60, No. 2 when averaged over power law size distributions, is the enhancement of the absorption. The imaginary part of the effective complex refractive index at the wavelength of 0.55*m was calculated for internally mixed, two-component aerosol models which contain these mixed particles as the absorbing component. Consequently it was found that the imaginary part of the refractive index for internally mixed aerosols can be larger by a factor of about 1.5 than that for externally mixed aerosols. However, twolayered particles with spherical symmetry as treated in this study will be an extreme case of the internal mixing of absorbing and nonabsorbing substances: in the real atmosphere more loosely bound mixed particles such as aggregates will exist. Therefore, the results obtained here should be considered to give an upper limit of the effect of internal mixing. Of course detailed study is needed on the effect of nonsphericity of actual aerosol particles before a firm conclusion can be reached. Together with our previous work (Mita and Isono, 1980), the following may be concluded. The absorptivity of atmospheric aerosols depends on the content, the size distribution, and the degree of internal mixing of highly absorbing substances such as soot and hematite. Though the degree of internal mixing is not well known for atmospheric aerosols, the effect of internal mixing is not very large. Therefore, the previous results obtained for externally mixed aerosols that the imaginary refractive index of atmosppheric aerosols should be about 10-2 in the visible region and that it rarely exceeds 0.05 will be true even in the case of internally mixed aerosols. Acknowledgements The author would like to express his sincere thanks to Prof. K. Isono who guided him into the field of aerosol and cloud physics. He also wishes to express his hearty thanks to Prof. A. Ono and Dr. H. Tanaka of Water Research Institute, Nagoya University, for valuable discussions and encouragement throughout this work. Thanks are also due to Prof. T. Takeda and Dr. Y. Iwasaka of the Institute for their encouragement. The author feels grateful to Prof. M. Tanaka, Tohoku University, for his useful comments on this work. The calculations were performed on the digital computer of the Nagoya University Computation Center. Appendix Derivation o f the expression (4) We start from the well-known Mie formula for the extinction cross section: where Re means taking the real part of its argument. For particles very small compared to the wavelength, it will be sufficient to consider only the first term in the series expansion. The coefficient a1 relevant to a coated sphere can be expressed as (Gi ttler (higher order terms), (A2) where i=*-1,q(= a/b) is the ratio of the core radius to the radius of the total sphere, and v is the size parameter of the total sphere as defined in the text of this paper. The coefficient b1 need not be considered here because it behave as ys, Using the above expression for a1, aext can be written as +(higher order terms), (A3) where I m means taking the imaginary part of its argument. The extinction generally consists of scattering and absorption. If the particle is nonabsorbing (i.e., both m1 and m2 are real), the leading term of expression (A3) disappears; this means that the leading term of the extinction expresses the contribution from absorption. Thus, substituting m1= n1- ik and m2 = n2-0 i into (A3) and after some manipulations, we finally obtain the following expression for the absorption cross section: where a = q*, and A, B, C, and D are constants given by Eq. (5) in section 3.3. A Appendix B Definition of the parameter The total volume Vc of the core material in a polydisperse system of mixed particles is given by p

11 April 1982 A. Mita 775 where n(a) is the size distribution function with n(a)da representing the number of mixed particles per unit volume of air whose core radii are between a and a + da; a1 and a2 are the lower and upper limit radii of the core. Similarly, the total volume Vs of the material forming the shell can be written as where *s(a) is the volume of the shell of a mixed particle having a core of radius a. *s(a) takes the following form: In the expression (B3) *1 and *2 are constants which do not depend on a. With Vc and VS siven above, the parameter p is defined as It is readily seen that p is related to k1 and k2 of the expression (B3) by p= k1*const (coast is determined by n(a), a1, and a2) and p=k2, respectively. The ratio of the outside radius of a mixed particle to its core radius, b/a, can be determined from the relation 4*bs/3=4*a3/ 3 + *s(a). This leads to In the present study, the sizes of the core of mixed particles (a) are assumed to follow the power law distribution. In case II the totalparticle sizes (b) also follow the power law distribution, since the ratio b/ a is independent of the particle size. In case I, on the other hand, the distribution of total-particle sizes becomes different from the power law. Aden, A. L., and M. Kerker, 1951: Scattering of electromagnetic waves from two concentric spheres. J. App!. Phys., 22, Dalzell, W. H., and A. F. Sarofim, 1969: Optical constants of soot and their application to heatflux calculations. Trans. ASME, Series C, J. Heat Transfer, 91, Espenscheid, W. F., E. Willis, E. Matijevic, and M. Kerker, 1965: Aerosol studies by light scattering. IV. Preparation and particle size distribution of aerosols consisting of concentric spheres. J. Colloid. Sci., 20, Fenn, R. W., and H. Oser, 1965: Scattering properties of concentric soot-water spheres for visible and infrared light. App!. Opt., 4, Finfer, E. Z., 1967: Fuel oil additives for controlling air contaminant emissions. J. Air Pollution Control Assoc., 17, Giittler, A., 1952: Die Miesche Theorie der Bengung lurch dielectrische Kugeln mit absorbierendem Kern and ihre Bedeutung fur Probleme der interstellaren Materie and des atmospharischen Aerosols. Ann. Phys., 11, Ivlev, L. S., and S. I. Popova, 1973: The complex refractive index of the matter of the disperse phase of the atmospheric aerosol. Izv. Acad. Sci. USSR Atmos. Oceanic Phys., 9, Kattawar, G. W., and D. A. Hood, 1976: Electromagnetic scattering from a spherical polydispersion of coated spheres. App!. Opt., 15, Kerker, M., J. P. Kratohvil, and E. Matijevic; 1962: Light scattering functions for concentric spheres. Total scattering coefficients, m1=2.1050, m2= J. Opt. Soc. Am., 52, Kerker, M., 1969: The Scattering o f Light and Other Electromagnetic Radiation. Academic Press, 666 pp. Klappenbach, E. W., and S. K. Goranson, 1979: Evaluation of Midwest sources of high particulate concentrations on October 15, 1976: a case study. J. Air Pollution Control Assoc., 29, Kondratyev, K. Ya., 0. B. Vassilyev, V. S. Grishechkin, and L. S. Ivlev, 1974: Spectral radiative flux divergence and its variability in the troposphere in the s region. Appl. Opt., 13, Kondratyev, K. Ya., R. M. Welch, 0. B. Vasilyev, V. F. Zhvalev, L. S. Ivlev, and V. F. Radionov, 1979: Calculations of free atmospheric shortwave spectral characteristics over the desert (from the CAENEX-70 data). Tellus, 31, Major, G., 1976: Effect of gases, aerosols and clouds on the atmospheric absorption of solar radiation. Beitr. Phys. Atmos., 49, Mita, A., and K. Isono, 1980: Effective complex refractive index of atmospheric aerosols containing absorbing substances. J. Meteor. Soc. Japan, Ser. II, 58, Prishivalko, A. P., and L. G. Astafyeva, 1974: Absorption, scattering and extinction of light by water-coated atmospheric particles. Izv. Acad. Sci. USSR Atmos. Oceanic Phys., 10, EZosen, H., A. D. A. Hansen, L. Gundel, and T. Novakov, 1978: Identification of the optically absorbing component in urban aerosols. App!. Opt., 17,

12 776 Journal of the Meteorological Society of Japan Vol. 60, No. 2 Volz, F. E., 1972: Infrared absorption by atmospheric aerosol substances. J. Geophys. Res., 77, Yamamoto, G., and M. Tanaka, 1972: Increase of global albedo due to air pollution. J. Atrnos. Sci., 29,