RL 682 PARTICLE SHAPES FOR MAXIMIZING LOW FREQUENCY ABSORPTION T.B.A. Senior and H. Weil Radiation Laboratory Department of Electrical and Computer Engineering The University of Michigan Ann Arbor, Michigan 48109 ABSTRACT The effects of particle shape and refractive index on the absorption efficiency of leigh particles is investigated analytically using spheroidal particles as prototypes. The effect of particle shape on absorption by Rayleigh particles is studied by ig spheroidal particles as prototypes. Since spheroidal Rayleigh particles can be fled analytically one can determine the relative absorption efficiency of clouds of ile-like, thin plate-like and spherical particles containing the same mass of particle volume of the cloud. We begin with the theoretical expression for Rayleigh scattering of a plane wave Einc = ei(k-r-owt) x irbitrary elliptical polarization a (a a*= 1) by a single homogeneous non-magnetizable;icle. The scattered field can be written as -Es eikr-iwt E = -e(r) %e - (kz)3 P_ (~xS = - r x (r x ) p is the electric dipole moment of the scatterer. In terms of the polarization tensor - = a RL-682 = RL-682 p = ~cP(e) * a Er is the particle dielectric constant relative to s0, the permittivity of the dding medium. For magnetizable materials S has an additional term involving the ced magnetic moment m which is expressible in terms of the same tensor as a function of eability instead of permittivity; i.e., P(u).

We'will obtain the absorption via the forward scattering theorem which states that ^* a x ( )] = -Im(a * S) ext k2 scatterers rotximation, vally id in the Rayleigh scattering approximations ext +a. hinduced magnetic dipole which may be found by a similar analysis. There are no terms1. For forward scations. Three types of averaging are r =. Hence ^ ^ ^ ^ A ^ = a = xl2 + P2am(2 yl + P331a z12) scatterers rotationally symmetric about the z axis entations.so we end up with a = kIm[Pll + P33 - Pll)Ja z2] For magnetizable particles, permeability p ~ p0' there are additional terms due to induced magnetic dipole which may be found by a similar analysis. There are no terms resenting magnetic dipole-electric dipole interaction. To make use of these results for particle distributions (clouds) we must average particle orientations. Three types of averaging are of interest: <la ~ z12>: Ha - ~ Im(2Pl, + P33); situation occurs for purely random particle orientations. <la ~ z12>: 0 aa k Im P1 1; situation could occur for example when the incident radiation is vertical and the ietry axes of the particles cluster about the vertical. This can be the case for ids of thin flat discs under some conditions; alternatively horizontal propagation,:ical polarization and horizontal needles.

- ^<a X3 1 Ca = 2 Im(P11 + P33) 3 situation would be generated, again assuming the incident radiation is vertical in action and plane polarized, by particles such as spindles whose symmetry axes are Jomly oriented in a horizontal plane, or by such particles not randomly oriented but idiated with circular polarization. The elements P1 and P33 for a homogeneous spheroid of volume V and complex lectric constant E, are (Ref. 1) 2 V1 Pll = 2V (q + ) -1 P33 = V _q + ) lengthr - '1 length z and width w 'e for a prolate spheroid of q 2 - -1I q = -[( )2 - l)1n +12+; = (Q/w)[(~/w)2 - 1]-1/2 an oblate spheroid q = C(C2 + 1) tan 1 - 2,e: = (/w)[1 - (Z/w)2]-1/2 i varies continuously a spindle (approximately from i = 0 for a very thin lozenge or disk through i = 1 an eyeless needle) to = o for a sphere. Writing cr = 1 + a + ib with a and b real and non-negative yields Im P1l = bV rl Im P33 = bV r3 2

~-1 pr = 4[(2 + aq)2 + (bq)2]j1 r3 = {[1 + a(1 - q)]2 + [b(l - q)]2} Then a = kVbr re, for the three types of averaging (a) r = (2rl + r3)/3 (b) r =, (c) r = (rl + r3)/2 s for a disk (q = 0) r = 1, r3 = [(1 + a)2 + b2]-1 for a needle (q = 1), rl = 4[(2 + a)2 + b2)]-1, r3 =1 a sphere (q = 2/3) r1 = 9[(3 + a)2 + b2]-1 = r3 that the same value for aa is found for each type of averaging. In Figure 1 we plot 3r vs. q for the two cases. a = 0.56, b = 0.83 implying n = 1.29 + iO.32 a = b = 3.0 implying n = 2.12 + i0.71. For type (a) averaging we see that for both materials, discs give the most )rption while spheres give the least. For type (c) averaging the spindles have a slight edge over discs in their )rption ability, and again spheres are poorest. For type (b) averaging both cases give the same result for disks and disks are;iderably more efficient absorbers than spheres or spindles. Thp tvnp (a) rpslilts have been given, by a somewhat different derivation, in 'f: 2.

If magnetic dipole contributions had been retained the results would be far more iplex since not only complex cr but complex r itself and the ratio v r/r would er as parameters. 'erences T.B.A. Senior, "Low-frequency scattering by a dielectric body," Radio Sci 11, 1976, pp. 447-482. T.B.A. Senior, "Effect of particle shape on low-frequency scattering," Appl. Opt. 19, No. 15, 1 August 1980, pp. 2483-2485.

3.0 b) 2.5. a = 0.56 (c) b = 0.83 2.0 3 (b) a = 0.56 b = 0.83 b =: 3.0 0.51 b = 3.0! t ~o1 t0 0.25 0.5 10.75 1.0O disc sphere spindle q 1: Particle shape influence on absorption for the three types of orientation averaging. (3r vs. q)