DIELECTRIC CONSTANT, ABSORPTION AND SCATTERING 275 
The Scattering Amplitudes 
a’ and b’, 
The scattered fields (E,, H,) outside the sphere 
and the transmitted fields (E,, H;) inside are due to 
forced oscillations of the sphere caused by the in- 
cident field (E;, H;). The fields (E,, H,) and (E;, 
H,) given by equations (7) and (8) can be regarded 
as due to electric and magnetic 2"-poles (n = 1 cor- 
responds to dipoles, n = 2 to quadrupoles, ete.) in- 
duced in the substance of the spherical particle. In 
the steady state these poles oscillate with the fre- 
quency of the incident radiation field. When this fre- 
quency approaches a characteristic frequency of the 
free vibrations of the electric or magnetic 2"-poles of 
the sphere, resonance will occur. It can, indeed, be 
shown that the amplitudes u, are associated with 
vibrations of magnetic poles and the b,’s with vibra- 
tions of electric poles. The characteristic frequencies 
of the free vibrations of magnetic poles of a sphere 
are determined by a condition which annuls the de- 
nominator of a,, those of electric poles by a condition 
which annuls the denominator of 6,, given by equation 
(14).78® The characteristic frequencies of the free 
vibrations are, however, complex in contrast with the 
real frequency of constraint of the radiation field 
falling on a sphere, as in the present case. The de- 
nominators of the amplitudes a, and 6,, although re- 
duced, can never become zero, and there are no diffi- 
culties caused by resonance. 
A glance at the formulas (14) shows the com- 
plexity of the amplitudes a, and 6,. An exact com- 
putation of these coefficients is out of the question on 
“account of the lack of tables of Bessel and Hankel 
functions of complex argument in the range needed 
here. They reduce to simple expressions in the limit 
when the parameter p = 2ra/A <1. In the present 
work we shall be mostly interested in the cases where 
p <1 or p <1. In these cases a series expansion of 
the amplitudes in ascending powers of the parameter 
p can be used. With the expansions of the spherical 
Bessel and Hankel functions 
Cj) A ee Gee as 
ae 2 am oan 
C) 
HOD) = eee DS 
m=0 
? 
(=e Cosa ae 
m!(2n+2m+1)!° 
j ~ (2n— 2m)! 
Niassa IO 2 
2 pn ti“, m!(n—-m )! 
+ 
used in equation (14), one is lead to the following 
amplitudes, keeping the first few terms of the ex- 
pansions and assuming p, = py, i.e., the equality of 
the permeabilities of the medium and the sphere: 
Since henceforth we will deal only with the scattering co- 
efficients a§, b’, we will omit the superscript s. 
nh — j2m( n! ye pints. 
(2n+1)!7 2n+3 
N?-—1 
1 2 pe +... 
[ +0(T— a) ip Ge) 
at 
b, =—jam( 4 
(2n + 1)! 
N*+1 
2n + 1) (n+ 1) (N?— 1) 
= as ep. ae) 
(2n + 1) [ (2n—1) N?—n—1] 
[ 1+ or Get On aad 5, 
— j2m( n ) 
(n+)! 
Gn DO EDAD ts] 
WN? eed & ne 
From these expressions one derives at once the ex- 
plicit formulas representing the induced magnetic 
dipole (a,), electric dipole (b,), and electric quad- 
tupole (b,) amplitudes. One has, then, neglecting 
powers of p higher than the sixth, 
ey aNee eal 3.N2—2 
iy ee ed lis 2 
3 N?+2 5 N?+2 
2j N?—1 :) 
ea 38)! 
5 42? , (38) 
and 
aaa Nora! ae 
1~ Gi ONeaoe 
It would appear interesting to present the relation- 
ships connecting the amplitudes of the electric and 
magnetic poles a, and 6, with those appearing in the 
treatment of Mie which was used by Ryde.*7 The 
magnetic and electric amplitudes in Mie’s notation 
are respectively p, and d,, and the relationships in 
question are the following: 
padic = (—)"j (2n + 1) an, 
a,Me = (—) "17 (2n+1)b,. (89) 
Finally the formulas (38) can be transformed so 
‘Some misprints and slight errors in the expressions for 
these amplitudes may be.noted in reference 18a. On page 571 
in the formula (35) and in the denominator of the coefficient 
of p%, read (2n + 2) instead of (2n + 1). In the formula (36) 
the minus sign on the right-hand side is missing. In reference 
18a, bi” and 627 have the wrong sign and the p® term in b;” 
is incomplete. Itis recalled that +7 has been replaced through- 
out this report by —j. 
