NEW TABLES FOR MIE SCATTERING FUNCTIONS FOR SPHERICAL PARTICLES, PT. 4 VALUES OF AMPLITUDE FUNCTIONS AM AND BM FOR REFRACTIVE INDEX N EQUALS 1.486
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Publication Date:
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STAT
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STAT
GEOPHYSICAL RESEARCH PAPERS
NO. 45
NEW TABLES OF MIE SCATTERING FUNCTIONS
FOR SPHERICAL PARTICLES
Part 4: Values of Ampi:tude Functions am and bm for Refractive Index n = 1.486
and for Size Parameters a = 0(0.1)30
Requests for additional 0000 by Agencies of the
Department of Defense, their contractors, and othergovernment agencies should be directed to the:
ARMED SERVICES TECHNICAL INFORMATION AGENC
DOCUMENT SERVICE CENTER,'
DAYTON 2, OHIO
Department of Defense contractors must be established
for ASTIA services, or have their 'need-toaknow' certi
fied by the
nizant military agency of their project
? or contract.
? ?
All other persons and organizations should app
U. S. DEPARTMENT OF COMMERCE
OFFICE OF TECHNICAL SERVICES,
WASHiNGTON 25, D. C.
MARCH 1956
GEOPHYSICS RESEARCH DIRECTORATE
AIR FORCE CAMBRIDGE RESEARCH CENTER
AIR RESEARCH AND DEVELOPMENT COMMAND
STAT
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Abstract
CONTENTS
d ? ? ? 11110?? ? It I ? ? 040 OOOOOOOO ?
1. The Problem
2. Mie's Theory
**I II* ?? 000 00 000 40 04 000 000 00 ??
0 0 ? 00 4 ? 0 II III ? ? ? 11 ? ? ? 0 ?? II 0 ? ? 4
3? Computations of the Amplituae Functions am and bm
4. Accuracy
? ?? ? 0 ? ? ? ? 440
00004 0 101408 000 00 04104 Se ? 44,11 4 GO* ?
5. Use of Tables
???????????????????????????????'?
References ? 04 004 041 00010100 0,0000*? 00 *OA 44
'fables of R(am), 1(am), R(bm), 1(bm), for a 0 (0.1)30
OOOOOOOOOO
4
6
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ABSTRACT
Tables are given for the real and imaginary parts of the complex amplitude functions am and bm.
They are the basic values for the computation of Mie scattering coefficients. The computations have
been carried out with the IBM 701 electronic data processing machine.
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r0.4
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I.
?
4
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1
1. The Problem
The scattering of light in the atmosphere has become of increasing interest and importance to
research in certain fields of physics, meteorology and colloid chemistry. A sound knowledge of the
scattering characteristics of particles of size comparable to the wavelength of the incident light is
necessary in studying the effects of light scattering.
This problem has been solved theoretically over forty-seven years ago by G. Miel. His
formulas are exact solutions of Maxwell's equations for spherical particles of any size and material.
He integrated Maxwell's equations for the electric and magnetic field inside and outside a particle and
calculated the amplitudes of the waves of light radiated by the particle in various directions. The exact
solutions appear in the form of series which converge more slowly with increasing radius of the particle
if the wavelength of incident light remains fixed.
Various investigators have since used his formulas to carry out numerical computations. Unfortu-
nately, this becomes an extremely time-consuming task for large radii due to the slow convergence of
the series. Some of the earlier computations are unreliable for large radii because not enough terms
were carried. As a rough approximation it may be said, that the number of terms in the series which are
required before truncation, is equal to the value of (X, the size parameter. Hence, the available data
are limited.
The only reliable results for large size parameters, that is, particles of large radius, are pub-
lished by Gumprecht and Sliepcevich2 . However, they used very large increments of the size parameter.
The existing tables do not meet the need we have for such tables; therefore, we planned and carried out
new computations.
2. Mies They
The diffraction theory undertakes to solve Maxwell's equation for a plane wave coming from
infinity hitting any surface of discontinuity of the optical parameter (index of refraction, n). An exact
solution can be found only for simple geometric surfaces of discontinuity, in such a case the equation
reduces to a boundary value problem for ordinary differential equations.
Miel solved the case for an incident plane wave falling upon a sphere of radius r. The sur-
rounding medium is transparent, homogeneous and isotropic. The incident wave induces forced oscil-
lations of free and bound charges which are synchronous with the applied field, inducing a secondary
electric and magnetic field each of which consists of two partsommely one inside the sphere and an-
other one outside the sphere. We ask for the intensity of the scattered light as a function of the scat-
tering angle 0. Thus, the intensity scattered, J, is a function of four parameters, r, X, 0, and n, namely.
J
Manuscript received for publication 20 June 1955
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Jo f jr, X, n,
(2.1)
2
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The solution has the form of infinite series, the coefficients of which are in turn determined
by infinite series of Bessel functions and Legendre polynomials.
We define the following terms:
n = index of refraction (of the aerosol particle against the dispersion medium)
r = radius of spherical particle (aerosol particle)
X= wavelength of incident light in? the surrounding medium
21t r/ X = size parameter
0= scattering angle (angle between the incident and the scattered beam)
B: = Cos
Plane of observation = plane containing the direction of the incident wave and the direction
of the scattered wave.
If the incident radiation is unpolarized (natural light) and its intensity is J0, then the intensity of the
radiation scattered in the direction is given by Mie as
(ii laP n7 GI i2
(X)11,
(2.2)
The quantities 111. and /2 are called intensity functions. They have the following meaning: The intensity
functions and 12 are proportional to the magnitude of the two incoherent, plane polarized components
scattered by one particle; 11 is proportional to the intensity whose electric oscillations are perpendicular
to the plane of obseirvation, i.e. polarized light in the horizontal plane; i2 is proportional to the intensity
whose electric oscillations are in the plane of observation, i.e. polarized in the vertical plane.
This result already shows that the scattered radiation is always polarized if the incident radiation
is unpolarized. The intensity functions* derived by Mie are
co
N-1 2m + 1
1 m (m+.1)
1
2m + 1
in (m+1)
]
a 'ft + b T
m m m m
[a Tm bm
2
2
?
(2.3) .
The vertical lines indicate that the absolute value of the complex argument is to be used. It is seen that
the scattered wave is composed of various partial waves and the amplitudes of these partial waves depend
on am and b The terms in these series are defined as follows. The amplitude functions am and bm are
complex numbers:
fol???????????1011141.??????????Voterayil 0.0.40.??????
? The notation used here differs somewhat from the notation used by Miel and those by Lowan3. It also differs
slightly from the first Mite on those studies written by one of us (B. Goldberg4). The formulae are written in
such a way as to make machine computations simpler.
.4
Si
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am
bm I (X, n = (-1
m+1/2 1131 sm 104
St. ip} (Dm To(1
+3/2 n S'm
Sm
{a}
st,
al sok P
101
Sm
S'm
c iz}
S'm
lz}
.{P} 1?111 la}
(-1)m
{z1
- 1/2)
= Sm + Cm zt
3 z
6`I'm Izt
The argument z takes on the values or in Eq. (2.4), and
m+1/2 = Bessel function of positive half order,
"111%/2 Besse' function of negative half order.
(2.4)
The amplitude functions am and bm can be interpreted physically as the in'th electric partial
wave and the rn'th magnetic partial wave, respectively. The solution (2.3) gives the scattered field as
a spherical wave composed of two groups of partial waves. The first group, the electric partial wave
am is that part of the solution for which the radial component of the magnetic vector in the incident
wave is zero. Similarly, the second group, the magnetic partial wave bm refers to zero radial compo-
nent of the electric vector.
The values of am and bm depend on the material (index of refraction n) and the size of the
particle (size parameter 00. The partial waves can be considered as coming from an electric or
magnetic multipole field. The first partial wave comes from a dipole, the second from a quadrupole
and so on.
The results show that the amplitude function am+1 of the (m+l)th electric partial wave is of
the same order of magnitude as bm of the m'th magnetic partial wave as already mentioned by Mie,
4nd can be checked by any table listing the values for am and bm (for example, Lowan).3 An ex-
ample may show this. For a 0.1 only 1 value of am and bm has the greatest order of magnitude
(101 and 102), which means for a= 1 the first partial wave dominates, and for a= 20 about 22
. -FDP8 1-01 043R002500180002-.
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partial waves have amplitudes of the same order of magnitude. The number of the partial waves ? neces-
sary for numerical computations ? increases with a, hi general, the number of terms in the series in-
creases with (X, and above a= 10 is given by
ma= 7 + 1.2 (X,
for our values computed for Ur-.0 to 30 and the Index of refraction in the range n 1.33 to 1.50. For a
< 10 the number of terms is
(2.6)
1 2 3 4 5 6 7 8 9 10
5 7 9 10 12 13 14 16 17 19.
The coefficients ffin and 'Cm are essentially angular functions containing the scattering angle 0, but
are completely independent of n and a:
dP.m,
irtm .1?I?1.8.1,1/.4
d x
d2 Pm
x2
- (1. 2) 7C'm,
(2.7)
where Pm is the Legendre polynomial of degree m. The functions Ttm and Tm involve the first and second
derivative of the Legendre polynomial of order m and argument z cos G. An example may show this. For
the third partial wave
45 3
15 2 ? _3 and ?3 ...- _2_ cos
cos ?
2
2
33
cos 8.
We shall discuss these functions further in part 7 of the publication where we compute the angular Mie
scattering coefficients.
3. Computations of Amplitude Functions am and bm
The computations for am and bm as can be seen from equations (2.4) involve the calculations of
of Sm 1 z} , Cm z} , S'm and C'm ./tz for the values of m required for convergence
of the series. In programming the computations, these functions were computed using recursion relation-
ships obtained from those satisfied by the Bessel functions:
z = sin z
sin z
Si z
Sm
and s'M
2m-1 ,
m
Sm.2
m Sm
Cm
= COS Z
cos z
+ sin z
2m-1
?
Cm.2 (3.1)
?z cm + Cmsi .
(3.2)
to
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?
Then we write
stm { [3}, s?;
stm cm
n
nS p} sm
st. (31 C.
la}
1'1
The amplitude functions Eq. (2.4) can now be written as
10(,n
or
am
(.1)m+1/2
S-Fit
m.
la,n )m.
(1)m
(-1)IM+1/2
s' 104 s. {P}
Sm ,
s'm la Sm ,
Cm {a} Sm
1 .
1 + (t/s)2 I
( 1 ::/(st/s)2
1
1 +-Tt-TsTr
tis
1 + (t/s)T
A similar derivation yields bm. Hence the amplitude functions are given by
am
bm
or in the standard terminology
(..)m (t/s)
- 1 + (t/s)2
(-1)m+1(v/u)
1 + (v/u)2
+
+ i
(..)m
1 + (t/s)2
(.)m+1
1 + (v/u)2
= R (am) + I (am) ,
= R (bm) + I (bm)
(3.3)
(3.4)
(3.5)
In computing the terms of the series, it is found that as a increases the number of terms required for
convergence increases. The terms oscillate about ?1 up to some m and then begin to decrease very rapidly.
In the recursion relationships for Sm and Cm for small z, accuracy is lost fairly quickly as m increases, but
since the number of terms required is small, this is of no consequence. For large z, where the number of
terms required is increased, there is again no difficulty because of the division by z. It was also found that
in computing s, .t, u and v, not much accuracy is lost due to the subtractions until the terms of the series be-
come sufficiently small. When this is the case the number of digits required in that term is not as great.
The derivatives were obtained using the previous functions rather than the previous derivatives so that the
error due to differentiation was made as small as possible. Our definition of am and bm (2.4) differs from
that used by Lowan and by Gumprecht and Sliepcevich by a factor
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2m + 1
m m + 1)
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This factor appears in our definition of i and i2 (2.3), since it is more advantageous from the com-
putational standpoint. Our and i2 values agree numerically with those in the other tables.
These computations were made using the IBM 701 Electronic Data processing Machine and were
programmed using IBM's floating point system, Speedcoding I. The sine and cosine subroutines used in
the computation of x cos ?, So and C give values which are good in the range considered, to eight
decimal places. The tolerance placed on 0 the truncation of the terms was le for both Real (am) and
Real (bm).
4. Lt.E.F2aci
The results for am and bm are believed to be valid to 6 significant figures. The results have
been spot checked against existing tables, like Lowan's3. and Gumprecht and Sliepcevich's2. The results
agree with those found in other tables.
To insure against machine errors, every calculation was duplicated and a check sum carried with
the calculation. The check sums were matched by machine and the calculation rejected on non-match.
Human errors in transcription were eliminated by machine-printing these tables by the IBM 701
and photo-offsetting the resulting master sheets. The results were printed out to 10 digits, but the last
three were cut out before the offset printing.
5. Use of Tables
The tables are arranged for increasing values of (X. Under each avalue all terms of the amplitude
functions are listed until all four functions R (am), I (am), R (bm) and. I (bm) are smaller than 1 x
i08. For small a, only two terms fulfill this condition. But for 0( 30, 43 terms are necessary.
The first column lists the number of the term. The second column gives R(am), the third 1(amb
the fourth R(bm) and the last I(bm). The numbers are printed to read 0.xxx, i.e. the decimal point
a2pars always before the first figure. It is followed by the exponent to the base 10. They are printed
as xxx, and -011 means 10" 1.
The first line under 0( 0.1 should read:
R(am)
+ 0.1914571 x 10-3
1(am)
- 0.3665583 x? 10-7
R(bm)
? 0.2663852 x 10-6
I(bm)
+ 0,7096110 x 10-13
The machine printed ten digits. The last three were covered before offset printing. Hence,
our tables show seven digits, the last one is not rounded off. Six digits are believed to be valid and the
seventh is carried in the table so that the reader can round off, if he wants six digits for further com-
putations.
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REFERENCES
1. 6Mie, G., "Beitraege zur Optik trueber Medien". Ann, Physik 25, 377 (1908).
2. Gumprecht, R.O. and C.M. Sliepcevich "Tables of light scattering functions for spherical particles",
Eng. Res. Inst., Univ. of Michigan (1951).
3. Lowan, A.N., "Tables of scattering functions for spherical particles". Nat. But. Stand, Apple Math.
Series 4 (1949).
4. Goldberg, B., "New computation of the Mie scattering functions for spherical particles". J. Opi.
' Soc. Am. 43, 1221 (1953).
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Tables of complex amplitude functions
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:14
R (am)
I (am)
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41?11111111101N.
mannalawirmininnemomaramomamommimillimmarlommi
411.1101.11.11?116.11SIMP/ON.~.1.M.Malr
R(bin) I(b1)
01;
02
011
02,
02.i
03
02j
a=0.1
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-.3463662590 -005 +1199695853 -010 +1133525480 -007 -1284880014 -015
a=0.3
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-2617017843 -004 +6848782398 -009 +1698876648 -006 -2886181867 -013
+6538900732 007 -4275722278 -014 -2986881053 -008 +8921458424 -01.7.
?
+1228571552 -001 -1509615954 -003 -2740270702 -003 +7509084086 -007-'
.m1095588896 -003 +1200315043 -007 +1243895227 -005 1547275336-Q11-
03i +4643624300 -006 -2156324664 -012 -1159783765-308 +1345098380 -017.
021
' 03
02
03,
04
10
a=0.5
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-3316678479 -003 +1100035734 -006 +5905361990 -005 -3487330023 -010
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+1607655303 -014 -2581442210 -008 -6663843682 -017
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m.
?
0
R (am)
0111111111Pla
01'
+6528912816
-001
02
-1747993411
-002
' 03
+2275844454
-004
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-006
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01 ,
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05
+4395044012
"007
01
'
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?
-1588674781
-001
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-003
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11
I
11 (am)
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1144841141140141141404111.1.14414411414444.401.1.
1(am)
04
105;
01
02
R(bm) 1(bm)
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50-Yr 2013/09/09: CIA-RDP81-n1n4nPnn9cnn1sznnnn n
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Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/09/09: CIA-RDP81-01043R002500180002-9
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