Dec., 1958
AN APPROXIMATION METHOD TO THE MIE THEORY FOR COLLOID SPHERES
1537
AN APPROXIMATION METHOD TO THE MIE THEORY FOR COLLOIDAL SPHERES BY RUDOLFB. PENNDORF Electronics Research Laboratory, Research and Advanced Development Division, A VCO Manufacturing Corporation, Boston 16, Mass. Received June 18. 1068
An approximation method is outlined for computing the total Mie scattering coefficient K in dis ersed s stems of light scattering spheres. The method is valid for any size ammeter, from small to very large spheres an$ any ariitrary real refractive inaex n _< 2. Analytical expressions are devefbped for the phase and the amplitude of K a t the maxima and minima. The constants in these analytical ex ressions are determined from existing data computed by the exact Mie theory. Finally, K values for any arbitrary size are dPetermined by a graphical interpolation rocem. This approximation method leads to an accuracy of A 2 to 3% . _for K as compared to results obtained by the Mie tgeory. Results are shown for selected refractive indices.
Introduction With steadily increasing interest in light scattering by colloidal spheres and its use as an important research tool, a sound knowledge of the scattering characteristics of particles of size comparable to the wave length of the incident light is necessary in studying the effects of light scattering. This problem was solved theoretically fifty years ago by G. Mie.’ His formulas are exact solutions of Maxwell% equation for scattering of electromagnetic radiation by spherical particles of any size and material. Unfortunately, this solution of the scattering problem consists of a sum of complicated terms involving spherical Bessel functions and associated Legendre polynomials and no simple analytical formulas have been obtained for these sums. To avoid the enormous time-consuming computations using the rigorous Mie theory, methods have been developed t o compute the total scattering coefficient K , the scattering cross section u or the specific turbidity by approximating the basic Mie functions. Although such an effort is desirable, the results obtained are not at all encouraging for large spheres, because all such approximations break down above a certain size parameter, normally a t the first maximum or first minimum of the major oscillations. 2,3 Each mathematical simplification of the spherical Bessel functions leads to a simplified physical model. Either certain types of waves are then neglected, or waves which have very long paths in the neighborhood of the sphere are omitted, or similar physical processes are dropped. What occurs for large a,however, is just the superposition of many types of waves, all very weak and none really predominant. This is perhaps unfortunate from the computational point of view, but it is the reality which we face. Therefore, we look for an entirely different approach. For most practical applications an accuracy of about f 2% for the total scattering coefficient K seems to be sufficient. That means a smoothed function consisting only of the major oscillations, but of no ripples, is sufficient for light scattering problems. If a higher accuracy is demanded there is no other choice but the exact computation. I n this paper, analytical expressions (1) Mie. Ann. Phyaik, [4]26, 377 (1908). (2) An account of such methods is given by C. J. Boukamp, Rep. Prow. Phya., 17,35 (1954). (3) W. Heller, J. Chern. Phya.. ‘26, 1258 (1957).
are derived for the extrema of the total scattering function, A graphical interpolation, of these computed values leads to scattering coefficients which agree for any size with rigorously computed values within i 2 or 3%. However, it is not permitted to extrapolate the expressions beyond the limits tested by rigorous computations, namely, for refractive indices n > 2. Total Mie Coefficient K and Normalized Size Parameter p.-Let us first define the basic parameters and list the symbols used on this paper. The total M i e scattering coeficient K is defined as the total Aux scattered by one particle in all directions divided by the flux incident on the geometrical cross section d. Consequently, the M i e scattering cross section is defined as u = rr2K (a,n)
(1)
K is a dimensionless coefficient, depending only on n and a. The size parameter is designated as a, where a = 2 ?r r/X = kr. Here r is the radius of the scattering particle, k the wave number in free space and X the wave length of the incident light. In this paper the normulized size parameter p = 2a ( n - 1 ) (2) will play an important role. For the refractive index the symbol n is used. The Lorentz-Lorenz term is M = (n2 - l)/(nz 2). K , denotes the K value at a maximum or minimum of the major oscillations, computed from the exact Mie theory. The subscript y in K , denotes an integer for the order of the extrema. For the parameters computed from our approximation method the following symbols are used: K denotes the total Mie scattering coefficient, R‘, the R value a t the maxima and R,“ the R value a t the minima. The subscript y has the same meaning as above. An example may illustrate the results of computations using the exact Mie formulas. Figure 1 shows the total scattering coefficient K for n = 1.50, as computed for (Y = 0.1 (0.1) 30 using the IBM 701 Data Processing Machine.* The curve shows distinctively four major oscillations of K with superimposed ripples. Van de Hulst6 found that for n near 1, the normalized size parameter p = 2a (n - 1 ) possesses a basic significance. This parameter equals the (4) R. Penndorf, J . Opt. SOC.Am., 47,1010 (1957).
+
(5) H. C.van de Hulst. “Light Scattering by Small Particles,” John Wile%& Sons, Inc., New York, N. Y.,1957,p. 172.
RUDOLFB. PENNDORF
1538
VOl. 62
1.5 0.5
~
*
~
~
/
5.0
4.0 3.0
k4 2.0 1.0
0
1
2
3
4
5
6
7
8
1 0 1 1
9
( p / 2 ) = a(n
- 1).
1 2 1 3 1 4 1 5
Fig. 2.-Envelopes and average K values for all computed K values in the range 1.33 _< n 5 1.50 as functions of the normalized size parameter. All the computed 1500 points fall between the two envelopes. A mean curve can be constructed. Note that the factor pj2 was used a8 abscissa.
phase shift which a light wave suffers while passing through the particle along the diameter, Le., through the center. He has further shown that for n < 2 the phase p of the K function agrees more or less for all 1 6 n 6 2, Le., the maxima and minima occur a t the same p value, independent of n. Hence, curves of K versus p are similar in phase. His findings are tested using our result^.^ Figure 2 presents the function K(p, n) for the range 1.33 6 n 6 1.50. A mean curve has been drawn through all the 1500 points and two envelopes are given. One envelope is drawn through the lowest K value for each p , another one through the highest K value for each p. The belt formed by the two envelopes shows the spread of the K values caused by the ripples. This spread of the K values due to the ripples is actually very small, being of the order of 0.1, except around the first maximum where it is somewhere larger, namely, about 0.15 unit in K. Only in the range p = 14 t o 19 are a few 0.3 unit larger than the average value K values raising the upper envelope which deviates there considerably from the average. This material for refractive indices proves that the normalized +ze parameter p is the correct and best way to describe the phase of the K function. Total Mie Coefficient K, at Extrema.-In this section analytical expressions will be derived for K a t the extrema, because the K function is com-
*
+
pletely determined if the phase (in terms of p ) and the amplitude (in terms of K ) of the scattering function a t the extrema are known as a function of n. First of all the phase (or position) of the extrema is investigated as function of p. Our data for n 1 and those for 1.33 6 n 6 1.50 have been c a r e fully analyzed. The phase p2/ denotes where K reaches a maximum or minimum. For n 1 a rigorous solution has been obtained6 for the phase of the extrema
-
.
-
py
=2
7ry
(3)
=F 2,
with lim xu +s/2
m
z - c m
For the limit eq. 3 leads to a constant value of the difference between two successive maxima or minima Apv = py
- pu-1
2r
(4)
For many practical applications this value 2 T is already reached after the first few extrema. I n this case eq. 3 assumes the simple solution py
+
=
2s, =F ?r/2 = 2r(y F
I/*)
(5)
where the sign applies to the phase of the maxima of the K functions and the - sign t o the minima. (6)
R.Penndorf, J. O p t . SOC.Am., 47, 603 (1957).
r
AN APPROXIMATION METHOD TO
Dec., 1958
THE
1530
MIETHEORY FOR COLLOID SPHERES
n
1
1
I
I
1
I
0.4
0.5
I
I
I
;7
5.0 .
1.33 1.2 9 1.20
.05 I .o
'
-0.2 -0.1
I
I
0
0.1
0.2
I
I
1111
0.3 I
I
O.BIO.91 1.01 1.15) 1.331 1.501 0.83 .93 1.05 1.20
0.6 0.7
n2-1 Ill(n2 t2)-
1.40 1.60
0.8 0.9 I.C
I
co'
2.0
n-
Fig. 4.-K, values for the first three maxima as function of M. The circles and crosses indicate the actual computed KYvalues. The scale is linear in M, but the scale for n is also indicated for the available n values.
0.05
o
0.1
-
0.1 5
I 0.. 'Y
0.20
0.25
For the first few maxima, the difference
Apv
-
11
+ 0.3 ( n - 1)
4 5.0 .6
21
PY.
~
3.0 2.0 1.5 1.0 0.5
(6)
N
Maxima
0 2 4 6
-
+
IG 2.5
where pa, { n 1] is taken from Table I. In summary, the data in Table I and the corrections expressed by eq. 6 suffice to ascertain the position of TABLE I POSITION pa, OF MAXIMAA N D MINIMAOF SCATTERING COEFFICIENT FOR n
0
in
listed in Table I, therefore, a small correction should be applied, because pv increases slightly with n. A satisfactory approximation is given by In) = pY In
THIRD MINIMUM
0.2 0.4 0.6 0.8 0.9 (n2 - l)/(nZ 2). Fig. 5.-KY for the first three minima as function of M. The and indicate the computed K u Values. The K," values for n 2 1 are represented by straight lines according to eq. 10.
4 i n In.rmr t , h m ~ / 2 mnre . sn fnr t,he mR.Yinis.
pa,
0
-0.2
Fig. 3.-x,' and E,'' as function of l / p y for selected refractive indices n. The solid lines represent 'eq. g and for selected refractive indices n. The vertical lines represent the normalized size parameter p, for = 1 to 6, according to Table I. en.
1 .z
8 10 12 14 16 18 20 22 24 26 28 30 a.
TOTAL MIE l.Oa
THE
Minima Pvi
1 4.0856 7.6231 2 10.7923 14.0041 3 17.1551 20.3266 4 23,4730 26.6312 5 29.7756 32.9278 6 36.0713 39.2202 7 42.3632 45.5101 8 48.6527 51.7984 9 54.9408 58.0856 10 61.2278 64.3720 a y is the order of the extrema, p u the position for the yth extremum, and xys = 2 r y - p , xyi = p - 2ay.
Fig. 6.-Smoothed I? functions for selected refractive indices n. The curves are based on the approximation method: ----, n = 1.10; - - - -, n = 1.20; - 0 -, n = 1.33; -X-, n = 1.486; ___ , n = 1.60.
the extrema, ie., correct phase of the R function for the whole range of n values, provided n 6 2.0. The next important parameter is the absolute value of K , at the-extrema, which determines the amplitude of the K function. Several approaches have been tried t o find a good analytical formula to cover the range 0.8 6 n 6 2.0. A novel approach, leading t o a better fit than the earlier attempts,T will be described. (7) R . Penndorf, Geophysical Research Paper no. 45, Part 6 (1955), p. 33.
RUDOLF B. PENNDORF
1540
VOl. 62
afunction. Equal lines of Aa are shown in steps of A a = 2. Some R functions are drawn in too.
Fig.7.-Three dimensional diagram of the
-
such a way that RlJ,agrees with K , for n 1 and n = m . The exact theoretical formulas (i.e., eq. 8 n 1 and the Mie formula for n = m are used. xu= 2 alpy b / p Y a c l - l / p , d f i f / p 2 (7) for This leads t o the following analytical expressions represents the amplitude of the smoothed K funcFor the maxima tion, ( i e . , without the ripples) best,; a, b, c and dare constants. The form of this analytical formula is 1, namely suggested by the exact formula for n 4 sin p 4(1 - cos p ) and for the minima K(n-1) =2--+-----
For the maxima and minima a relationship of the type
+
+
+
+
-
1
-
P
P=
(8)
A physical interpretation of eq. 7 can now be given.
K",
2
4
4
8.01M
--+ p u 2 + p y - PV
27.3M PU2
(10)
The first term represents the limiting value t o which K converges for large size parameters. The second term corresponds to the second term in eq. 8. Inserting p, from eq. 5 into the second term of eq. 8 leads to
I n summary, the amplitude of the X function can be computed fairly accurately from eq. 9, 10 foy a smoot_hed K function of any arbitrary n. K', and K", are computed according t o eq. 9 and 10 and for some selected n values shown in Fig. 3 as 4 for the maxima, and function of l/p. Vertical linea indicate the position -- sin pu = + PU PI of py. For n close to one, the lines are fairly straight, but with increasing n the curvature is - 4sin2! - -4 for the minima more and more pronounced. PY Pu To facilitate computation for any arbitrary n, because sin [2 r ( y - l / * ) ] = -1 and sin [2r(y l/4)] = +1, respectively. Hence a = f 4, Table I1 lists the numerical constants obtained where the sign applies to the maxima and the from eq. 9 and 10 for the first 6 maxima and minima. - sign t o the minima. The third term in eq. 7 corresponds t o the third TABLE I1 term in eq. 8 with b = 4, because cos [27r(y NUMERICAL CONSTANTS IN THE ANALYTICALFORMULAS 1/4)] = 0. This third term decreases very fast (EQ.9 AND 10 FOR THIOAMPLITUDE OF THE EXTREMA (1 5 y with increasing y, and canbe neglected for y > 5, -< 6); SMOOTHED SCATTEKINQ COEFF~CIENT ff when its contribution to K amounts to less than At the maxima Minima 1%. I?,' = 3.173 + 4.02M El" = 1.542 + 0.579M The last two terms in eq. 7 are the appropriate zzf = 2.404 + 2.25M &" = 1.734 + 0.433M corrections for n > 1, where the influence of the reRat = 2.247 + 1.52M Raw= 1.813 + 0.328M fractive index n is expressed by the Lorentz&' = 2.178 + 1.14M E," 1.855 + 0.262M Lorenz term M = (nz - l ) / ( n z 2). The con&' = 2.139 + 0.92M &" = 1.882 + 0.218M stants c and d for the maxima are determined by an Re' = 2.114 + 0.77M Ea'' 1.901 + 0.188M iteration method to fit best the smoothed K , values for refraction indices computed by the Mie theory, To prove the quality of these constants, Fig. 4 and the minima also by an iteration method, but in shows the K , values as computed by the Mie theory
+
+
+
+
