013714-1-F THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING Radiation Laboratory FINAL REPORT Grant No. ENG 75-03880, 15 April 1975 - 15 April 1976 Thomas B.A. Senior Prepared for National Science Foundation Washington, D.C. 20550,13714-1 -F = RL-2263 u/ 13714-1-F = RL-2263 Ann Arbor, Michigan

TIlE UNIVERSITY OF MICHIGAN RADIATION LABORATORY 013714-1-F SCATTERING BY SMAT.T AKROSOT.S This is the final report on NSF Grant E;NG 75-03880 entitled "Scattering by Small Acrosols" and covcrs thc period 15 April 1975 - 15 April 1976. The investigation was concerned with the electromagnetic scattering characteristics of diclectric crystals such as those occurring in ice crystal clouds and other aerosol-laden atmosphares. Whereas most current theories have modelled the crystals as spheres, our interest is centered on the computation of the scattering behavior of thc specific crystal shapes which arc known to occur. A portion of the frequency spectrum where this task is feasible is the so-called "low frequency" or Raylcigh region where the wavelength of the radiation is greater by an order of magnitude or more than the maximum linear dimension of the scatterer. The particles of interest consist mainly of plate and columnar crystals whose linear dimensions range from 10 to 500/.m. For even the largest of these crystals, the Rayleigh region encompasses the entire range of radio frequencies, with wavelengths dlown to the millimeter range, and for the smaller particles extends into the far infrared as well. Our studies have bcen directed entirely at the scattering from a single homogeneous isotropic dielectric particle at these frequencies. A general mathenmatical formulation has been derived to specify thle scattering behavior of such a particle and has been applied to the special cases of rotationally synmmetric bodies and rectangular parallelcpipeds. Techniques resulting from recent applications of digital methods in potential theory to electromagnetic problems have been utilized to obtain efficient numerical methods for computing the dipole moments which characterize the scattering, and data obtained for selected geometries. Formulation Tn our general formulation we have adopted an approach based on the electric and magnetic polarizability tensors P(er) and M(, ) respectively, in tcrms of which the electric and mnagnetic dipole moments which characterize the far zone 1

scattered fields arc simply the dot products of the polarizability tensors with the incident field polarization vector. A particularly convenient aspect of this approach is that the tensors arc functions only of the material and geometry of the scatterer and arc independent of the direction and polarization vectors of the fields. This explicit separation of the direction and polarization from the intrinsic properties of the partiele makes the formulation highly advantageous for any subsequent development of a multiple scattering theory by a cloud of particles. The formulation is described in detail itn attachiment A where it is shown that both tensors arc special cases of a general polarizability tensor X(7), where 7r is a material parameter representing either the relative permittivity r or relative permcability 1l of the particle. The tensor clements Xi (i, j = 1, 2, 3) are expressed as weighted surface integrals of certain potential functions or alternatively as integrals of the normal derivatives of the potentials. To compute the tensor elements it is therefore sufficient to determine either the potentials or their normal derivatives at the surface of the body, and integral equations have been derived for both of these quantities. Examinatiorl of these integral equations shows that unique solutions exist for T > 0, so that the tensor x(7) can be uniquely determined for the values of T which arc of practical interest, and requires the solution of at most three integral equations. Properties of x(7) Since the essential ingredient in the solution of the scattering problem is now the determlination of the general polarizability tensor, we have explored the mathematical properties of X(T). For real r, the tensor is real and symmetric; and for a body which is rotationally symmetric about the x3 axis, or indeed for any body having symmetry about the two perpendicular planes x1 = 0,2 = 0 and the plane x1 = x2, the tensor is diagonal with xl1 = x22, having therefore at most two independent clements and requiring the solution of (at most) two corresponding integral equations. In the case of a spheroidal body we have proved the relation X11(7) =2X33 ) (1) 2

holding for all spheroids, both oblate and prolatc. This implies that the conputation of x33 for O < T <oo yields the values of x l(= x 2) for - 1 < < co and obviates the solution of a second integral equation. For the general case of a body of arbitrary shape, it is possible to obtain bounds of a gcolmetrical nature on the diagonal elements of the tensor. A lower bound has been established as (c.f. eq. (28) attachment A) 7-1 (2) Xii. > V (2) where V is the volumle of the body, and we have shown as well that an upper bound is given by x < (T- 1)V. (3) These bounds arc valid for all homogcneous dielectric bodies having 7 > 0 and are optimum in the sense that the equality is obtained in (2) by the elements x 3 for a vanishingly thin oblate spheroid and X for a disc and in (3) by the element X33 for a disc. We also remark that the present results have indicated that more stringent bounds may be obtained in such special cases as, for cxample, rotational symmetry, but this has not yet been fully investigated. Applications The formulation has been applied to a rotationally symnretric body of homnogeneous isotropic material, and integral equations have been developed for this case in terms of boththe potentials and their normal derivatives at the surface of the body. Comrputer programs have been written for the numerical solution of the integral equations, and hence the computation of x11 and X33, and their validity has been checked by comparison with the results obtained from the known analytical expressions for a sphere. The programs are applicable to any rotationally synmmetric body whose profiale can be construced from straight line and circular arc segments. Data have been obtained for bodies of various shapes, e.g. spheroids, ogives and right circular cylinders, and although the relation (1) has been proved analytically only for spheroids, these data have indicated that (1) may 3

be valid for rotationally symrmctric bodies in general. The dipole momentrs per unit volume for an ogive arc closely approximated by those of the corresponding spheroid of the salme length-to-width ratio alnd mnaterial paramtcler T, the difference being about one percent or less, and for all practical purposes an ogive may therefore be represented by a spheroid. In addition, we remark that the formulation in terms of the potentials is far superior to that in terms of their normal derivatives from the standpoint of numerical convergence. We have also examined the case of a homogeneous isotropic rectangular parallelepiped. The basic integral equations of attachment A have been specialized for this geometry with the result (c. f. eq. (16) attachment B) 3 1 4 i (r) - 2X + 27 '.(r )K. (rr )dS (4) n = 1 n] and we remark that although the integral equations are in general weakly singular, the kernels K. in (4) are bounded provided the observation point (whose position 'Il vector is r) does not lie on an "edge" of the body. The numerical solution of (4) by the moment mnethod can be achieved by dividing the surfaces of integration into sampling cells over which the entire integrand in (4) is assumed constant. Comiputer programs have been written to compute X1 (= x22) and X33 by this method for a rectangular parallelpiped of square cross section, and data obtained for the special case of a cube. The results are presented in attachment B and differ substantially from those previously reported in the literature. For rectangular parall1lcpipeds of large length-to-width ratio the numcrical convcrgence properties of this method are rather poor, and the large number of sampling cells required to obtain accurate results greatly increases the expense of the computations. Since the kernels in (4) may be integrated analytically, we actually need only assulme the potential "Y to be constant over the surrace of a cell, and a second program has been written to compute the tensor clements by this method. A comparison of the two programs has been made using the cubic geometry and shows the latter to be far superior numerically, yielding greater accuracy with 48 cells than was 4

previously obtained using 75 cells. In addition to reducing the required number of sampling cells, we have also found that for bodies of large length-to-width ration, where the use of square cells is no longer feasible, the task of scleqtitg the configuration of sampling cells is simplified in the second method since we arc there concerned only with the potential, which is in gcenral a slowly varying function over the surface of the body. A third geometry which we have examined is a cylinder of hexagonal cross section. The mathematical formulation has been derived, and since it does not differ substantially from that for a rectangular parallelepiped, the development of numerical techniques for computing the tensor elements has been postponed pending further investigation of the methods described above. A proposal seeking support for the continuation of this work was submitted to the National Science Foundation on 25 March 1976. 5

Publications Supported by the Grant Senior, T.B.A. (1976), "Low frcquency scattering by a diclectric body, " to appear in Radio Sci. 11(5), 477-482 (attachment A). TTcerrick, D. F. and T.B.A. Senior (1976), "The dipolc moments of a dielectric cube," submitted to IE.PIE Trans. Antenna and Propagation (attachment 3). Internal Memnoranda Senior, T. B.A., "Low frequency scattering by a dielectric ellipsoid, " Memo 013714-501 (20 July 1975). Senior, T. B.A., "ILow frequency scattering by a homogencous dielectric body, " Memo 013714-502 (29 July 1975). Senior, T.B.A., "Computed low frequency scattering by a dielectric body," Memo 013714-503 (6 August 1975). Senior, T.B.A., "Correction of an error, " Memo 013714-504 (22 August 1975). Senior, T. B.A., "Low frequency scattering data for dielectric bodies, " Memo 013714-505 (3 September 1975). Senior, T.B.A., "Upper and lower bounds, " Memo 013714-506 (5 September 1975). Senior, T.B.A., "The polarizability tensor X," Memo 013714-507 (10 September 1975). Herrick, D. F., "Rectangular parallelepiped, " Memno 013714-508 (18 November 1975). Senior, T.B.A., "Some fundamental inequalities, " Memo 013714-509 (28 February 1976). Herrick, 1). F. "Solutions of the integral cquations, " Memo 013714-510 (26 March 1976). Adninistrative Apart from the project director, the only personnel receiving support from the Grant were Mr. D. iF. llcrrick (a research student), programmers and a secretary. 6