342 L E T T E R S T O T H E E D I T O R Vol. 52

Early History of the Mie Solution NELSON A. LOGAN

Lockheed Aircraft Corporation, Missiles and Space Division, Sunnyvale, California

(Received January 6, 1961)

IN recent years, scores of papers have been published which have dealt with the evaluation of the scattering properties of

spheres as a function of both the size and the electrical properties of the scatterer. Almost without exception, the basic formulas are referred to as the "Mie solution," "Mie formulas," or "Mie theory." The solution is often derived from the boundary-value problem for Maxwell's equations by using the Debye potentials which are said to have been first introduced by P. J. Debye in a paper published in 1909.1

The basis for the association of these formulas with the name of the German physicist Gustav Mie is a classic paper published

in 1908.2 Mie developed the complete analytical solution and discussed its significance in terms of partial electric and magnetic waves. The solution was used as the basis for some numerical and graphical results related to the explanation of the colors of metal sols.

With the exception of a paper by Hawksley3 the present writer knows of no modern writer who acknowledges the fact that the so-called Mie solution was already well known when Mie and Debye published their classic papers.

An examination of the literature of the late l800's reveals that the formulas which came to be known as the Mie solution were first presented in a memoir published by the Danish physicist Ludwig Lorenz in 1890.4 Lorenz solved the vector-wave equation by using a pair of potential functions which are identical with the Debye potentials. One of the factors which contributed to the unjust neglect of Lorenz's paper by his contemporaries was certainly the fact that Lorenz advocated (up until his death in 1891) a theory of light which was at variance with the Maxwell theory. Although the physical interpretations in the theories of Maxwell and Lorenz were, radically different, the mathematical description of propagation and scattering of the "disturbance" are identical.

In 1893, the English physicist, J. J. Thomson,5 published the Mie formulas for a perfectly conducting sphere. Thomson did not use the Lorenz-Debye potentials because he used a solution of the vector-wave equation (for the Cartesian components of the vector-wave function) which had been given in 1881 by H. Lamb.6

In his 1908 paper, Mie discusses Thomson's solution for the perfect conductor. He also refers to Lorenz's theory for the color of metal sols made up of small particles. However, although Mie was familiar with Lorenz's preliminary papers, he was apparently unaware of the brilliant 1890 memoir.

In 1899, both A. E. H. Love and G. N. Walker8 realized that the Lamb-Thomson solution for the perfect conductor could be readily extended to the case of a dielectric sphere.

Most modem authors seem to be aware of the fact that the solution of the boundary-value problem for the scattering of scalar (acoustic) waves by a sphere was first given by Rayleigh9

in 1872. However, it is apparently not well known that Lamb10

acknowledged that his vector-wave solution of 1881 (or, its equivalent, the Lorenz-Debye potentials) had been discovered by Alfred Clebsch as early as 1861." Clebsch showed how to solve exactly, for any ratio of size to wavelength, the problem of determining what, in modern vector terminology, is known as the dyadic Green's function for the elastic wave equation for the case of a perfectly rigid sphere. This was a remarkable feat since Clebsch had to work with the "long-hand" notation used before the development of vector analysis.

This history should serve as a caution to modern writers who should endeavor to establish that new computations which they publish are truly new.

The above historical remarks are a condensation of a detailed history of these problems which will be presented in a report which is now in preparation.12

1 P. Debye, Ann. Physik 30, 57-136 (1909). 2 G. Mie, Ann. Physik 25, 377-442 (1908). 3 P. G. W. Hawksley, Monthly Bull. Brit. Coal Utilization Research

Assoc. 16, 117-147, 181-209 (1952). 4 L. Lorenz, Videnskab. Selskabs Skrifter 6, 1890; Oeuvres Scientifiques

I, 405-502 (1898). * J. J. Thomson, Recent Researches in Electricity and Magnetism (Oxford

University Press, Oxford, England, 1893). (See Sec. 369-379 entitled "On the scattering of electric waves by metallic spheres.")

6 H. Lamb, Proc. London Math. Soc. 13, 189-212 (1881). 7 A. E. H. Love, Proc. London Math. Soc. 30, 308-321 (1899), 8 G. N. Walker, Quarterly Journal 31, 36-49 (1900). 9 Lord Rayleigh, Proc. London Math. Soc. 4, 253-283 (1872). 10 H. Lamb, Hydrodynamics (Cambridge University Press, Cambridge,

England, 1932), 6th ed. (See footnote on page 636.) 11 A. Clebsch, Z. Math. (Crelle's Journal) 60, 195-262 (1863) (completed

October 30, 196l). 12 N. A. Logan, "General research in diffraction theory, Vol. III Asymp­

totic expansions for exact solutions for diffraction by circular cylinders and spheres," LMSD Report No. 288089 (Lockheed Missiles and Space Division, in preparation).

March 1962 L E T T E R S T O T H E E D I T O R 3 4 3