A simple method for determining the size of a sphere from the extrema of the scattering intensities. III. Conducting sphere

A Simple Method for Determining the Size of a Sphere from the Extrema of the Scattering Intensities III. Conducting Sphere A. j. PATITSAS Department of Physics, Fine Partlde Research Institute, Zaurentian University, Sudbury, Ontario, Canada Received July 23, 1973; accepted July 30, 1973 I t has been shown that a simple relation exists between the diameter of a conducting sphere with a given complex refractive index and the angular positions of the extrema of the Mie scattering intensities. This relation allows one to determine the positions of the extrema for a given diameter, or the reverse, without computational aids. The real pait of the refractive index was varied between 1.10 and 2.10 while the imaginary part assumed practically all values. The size parameter of the sphere varied between 4 and 24, but for relatively small scattering angles, or not very small imaginary part of the refractive index, the method is applicable for size parameters larger than 24.

I. INTRODUCTION This is the fourth part of the study recently undertaken to determine the diameter of a sphere of arbitrary refractive index from the extrema of the scattering intensities without computational aids. The first part (1) dealt with a perfectly conducting sphere of arbitrary diameter, the second part (2) dealt with a dielectric sphere of relatively small diameter and refractive index, while the third part (3) dealt with a dielectric sphere of large diameter or large refractive index. This work has its origin mainly in the researches of Dandliker (4), Maron and Elder (5), and Kerker el al. (6). An extensive discussion on the subject may be found in Chap. 7 of the book by Kerker (7). The procedure begins by writing ki for minima af(O0 =

Ki for maxima,

where a = reD~X, and f(0~) is function of 0¢. It equals 0~/2 in the case of the dielectric sphere when 0~ is measured from the forward

direction, and sin(01/2) when 0i is measured from the backward direction. It equals ½[-0¢/2 + sin (0i/2) -] in the case of the perfectly conducting sphere with 0i measured from the forward direction. The angle 0i, measured in radians, represents the angular position of the ith extremum. As usual, D is the diameter of the sphere and X the wavelength of the incident radiation. The procedure is completed when the parameters kl, Ki are determined as functions of the size parameter a and the refractive index N of the sphere. This determination is rather simple in the case of her perfectly conducting sphere (1) and in the case of the dielectric sphere with rather large diameter and with 0i measured from the backward direction (3). It is not so simple in the case of the dielectric sphere with 0i measured from the forward direction (2, 3). Work is now under way to extend the method to scattering of scalar waves by spherical well potentials in the quantum mechanics sense, and to scattering of acoustic waves by soft and

276 Journal of Colloid and Interface Science, Vol. 46, No. 2, February 1974

Copyright ~ 1974 by Academic Press, Inc. All rights of reproduction in any form reserved.

SIZE FROM LIGHT SCATTERING

277

14.0 " ' " .--,... .... 12.0

'...

"".-,..... ,.:. -":.,......

....:

I0,0

.... .......... ....

~JN Z

8.0

' " ' ~ .:.. ....

•.•_..•

-

+ ~1~ --{cu

6.0

..

4.0

• • .-...~...

.... _ . ~ _......:.:..:._.:~.~...,..-: 2.0

0 ~/ N" (I.500, 0.050) 4

8

12

16

N= (I.800, O. I10) 20

OC

4

8

12

16

20

OC

FIG. 1. The quantity a½ [-0d2 + sin (0i/2)] is plotted versus the size paramet a for two values of the complexrefractive index N. The angles 0l are the positions of the maxima of the perpendicular scattering intensity ix .The sloping lines act as boundaries between the two regions. The vertical bars indicate the values of a where jumps in the positions of the extrema occur. hard spheres. Effects of polydispersity will be considered shortly. II. METHOD In the computation of the Mie scattering intensities the real part of the refractive index N, designated by NR, was changed from 1.100 to 2.100 in steps of 0.100. The imaginary part of N, designated by NI, was changed from 0.010 to 0.220 in steps which became increasingly larger as N I became larger in an arithmetic progression like manner, the ratio being 0.010. Thus the first step was 0.010, the second 0.020 and the sixth 0.060. In some computations N I was changed at equal intervals. Also the cases with N R equal to 1.250, 1.330 and 1.350 were considered. The size parameter a was changed from 4 to 24 in steps of 0.20 and the scattering angle 0 was changed from 5 to 90 ° in steps of 0.4 °. In some computations the angle 0 was changed

from 5 to 160 °. Two typical outputs corresponding to the maxima of the perpendicular scattering intensity is, and for 0 in the range 5 ° < 0 < 900, are shown in Fig. 1 for N R = 1.500, N I = 0.050, and for N R = 1.800, N I = 0.110. In those outputs, as in the case of the perfectly conducting sphere (1), the average of 04/2 and sin(0i/2) multiplied by a is plotted versus a. Similar plots corresponding to the minima of il and also corresponding to the extrema of the parallel scattering intensity i2 and the average of is and i2 were produced. The main observation obtained from the plots in Fig. 1 is that the plane m a y be divided in two regions; the upper left region where the patterns of the points resemble those for the dielectric sphere (2), and the lower right region where the patterns resemble those for the perfectly conducting sphere (1). Examination of all outputs revealed that the two regions m a y be separated by a straight line

J o u r n a l of Colloid aria Interface Science,

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46,

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1974

278

PATITSAS

segment (Fig. 1) whose slope and vertical intercept at a = 24 increase with NI. After considerable persistence it became possible to arrive at the following expressions for the slope C and the intercept B at a -- 24, C = 5.30(NI) + 0.08 for

1.10
2.10

B = 96.67(NI) + d

[-11 [-21

where 2.53 -- 7.50(1.50 -- NR) for

NR<

1.50

for

1.50 < N R < 2.10.

d-2.53

It can be seen from Eq. [21 and from Fig. 1 that for N I > 0.22 all points lie in the lower right region. This was the reason for not considering in detail the cases with N I > 0.220. Once the separation of the plane into the two regions is effected, for a given N, it remains to apply the method in each region. a. Upper Left Region

In applying the method it m a y be thought that this region covers the whole plane. Then the procedure outlined previously (2) m a y be applied. The working equation is, ki = - - S n a + b + (n -- 1)A,

[-3~

Ki n = i+m--

1,

where S~ is the slope of the nth line, counted from the bottom, which best fits a given set of points (Fig. 1), b is the vertical intercept at a = 0 of the first line corresponding to the first extremum, A is the distance between adjacent vertical intercepts, and m is the index which enumerates the interval in which a lies. Thus, for the case N = (1.500, 0.1350) shown in Fig. 1, m = 1 if a lies in the range 0
occur as explained earlier (2). Thus, in the region to the left of the vertical bar at ~ -- 8.6 the lower set of points correspond to the first maximum, the next set correspond to the second maximum, etc. In the region between the vertical bars at ~ equal 8.6 and 19.2, the first set of points correspond to the first maximum, the second set to the second maximum, and the last set at the top correspond to the eighth maximum. I t can be seen that the set of points corresponding to the third maximum continues below the line separating the upper left and the lower right regions. This shows that for points near the separation line the error in the predicted values of the sphere diameter may exceed the guidelines given below. In particular the error for these points below the separation line is about 12% as the points are grouped together with the set of points corresponding to the third maximum in the lower right region. As in the previous case (2) the neighborhood 23 where the method is not applicable and which is centered about the points of discontinuities in the positions of the extrema, is about 0.6. The values a where the discontinuities occur are plotted versus N I in Figs. 2, 3 and 4 for the various values of N R listed above and also shown in the figures. The uncoordinated variation of the points of discontinuity as N I changes does not allow any practical analytical representation of the curves. The next step towards determining the parameters kl, Ki is to determine the slope S~. A convenient analytic representation in the present case is as follows, Sn = S -

[0.01(NR-

1) + 0.05(NI)]

( n - 1), [4~ n=

i+m--

1,

where S is the slope of the first line and is given as, 0.233(NR) -- 0.259 S=

+t-2.800(0.01 -- NI) 0.233(NR) -- 0.259

for N R < 1.20

Journal of Colloid and Interface Science, Vol. 46, No. 2, February 1974

for

NI_< 0.01

for

N I >_0.01,

SIZE FROM LIGHT SCATTERING

//,2,

24

,,,6

i.5

1.2

279

"

•

1.4

22

2018

"

ill

O

VQ,.

I.'

Z

12

z5

Jo

Vz

°

8

<

6

__ f

" --2.1

--I,8 4

"

0

FIRST I 0

.2.0

.04

DISCONTINUITY I I .08

.12

I

SEcoND t

.16

.04

DISCONTINUITY I I .08

I

.12

NI

NI

FIG. 2. The values of a where the first and second jumps in the angular positions Oi of the extrema of 11 occur are plotted versus the imaginary part of the refractive index for various values of the real part of the refractive index.

and

analytic representation, 0.660(NR) -- 0.784 +2.800(0.01 -- NI)

2x= for

NI_< 0.01

for

N I > 0.01,

1.57 -- 2.00(NI)

for

O.Ol _< N1 < 0.03

1.51

for

N I > 0.03,

S-0.660(NR) -- 0.784

for N R _< 1.30, and

for N R > 1.20. The intercept b of the first line at a = 0 is given exactly as in the case of the dielectric sphere (2), that is,

1.57 -- 2.00(NI) + ( N R -- 1.30)0.30 for

0.01 < N I _< 0.03

A= brain =

2.20 + 4S

for 1.10 _ N R <_ 1.14

2.10 + 4S

for 1.14 < N R _< 1.40

1.51 + ( N R -- 1.30)0.30

for

-- 2.00(NR) + 4.95 + 4S for 1 . 4 0 < N R _ < bmax =

2.10,

forNR>

NI_> 0.03

1.30.

bmln + 0.60.

The parameter h assumes the following

The values of A for N I in the range 0 < N I _< 0.01 m a y be obtained by assuming that it

Journal of Colloid and Interface Science, Vol. 46, No. 2, F e b r u a r y 1974

280

PATITSAS

'. ;,/,.,

24

22

J 7"

20

18

16

o I-

14

o z

12

I-

I0

I.-. z

8¢.0

19 THIRD

DISCONTINUITY

FOURTH

DISCONTINUITY

6

=.. ,.,.1 4

2

0 0

,04

.08

.12

.16

N[

.04

.12

.08 N I

I

Fro. 3. Same as in Fig. 2 except that the thiId and fourth jumps are considered. varies linearly from its value at N I = 0.01 to its value at N I = 0.0 as determined in the previous paper (2). The error in computing kl, K i is about 15% when n = 1. I t becomes about 5 % when n = 5, and about 3 % when n = 10. When 0i > 90 ° the error m a y be considerably larger than these estimates

b. Lower Right Region This case is relatively easy to treat since there are no discontinuities in the positions of the extrema and the points corresponding to a given extremum tend to lie on straight lines of zero slope as in the case of the perfectly conducting sphere (1). I t can be seen in Fig. 1 that the line which best fits the points corresponding to the first m a x i m u m has vertical intercept at 2.43, while the intercepts of the

other lines can be obtained by adding successively 1.50. The same holds for the case of the minima except that the intercept of the first line is at 1.80. Thus, it is possible to write quite simply,

2.4o +

(i -

1)1.50

for maxima

a l F Oi

= 1.8o +

(i -

1)1.50

for minima.

[-5-]

This equation is valid for all N R , N I considered , i.e., 1.10 < N R _< 2.10, and 0.010 _ 24 provided the point {~, ~½F(0i/2) + sin(0i/2)-]} remains in the lower right region. The expected

Journal of Colloid and Interface Science, Vol. 46, No. 2, February 1974

SIZE FROM LIGHT SCATTERING

281

24

22

2O =E

SIXTH DISCONTINUITY

FIFTH DISCONTINUITY

o I-o o.

14

z

12

~z F-

I0

4

0

k

I

I

I

.04

.OEI

.12

.16

I

L

.04

.08 NI

NI

i .12

FIG. 4. Same as in Fig. 2 except that the fifth and sixth jumps are considered.

error is about 5 % for i = 1. I t is reduced to 1 about 3 % for i = 4. The same equation is also valid in the case of the parallel scattering intensity i2 and the average of il and is in a lower right region which is practically the same as that for il defined by Eqs. [-11 and [-2~. Calculations with N R = 1.40 and N I changing from 0.220 to 50.00 have revealed that Eq. E51 remains the same regarding il even for N I = 20.00, while for N I = 50.00 it is slightly modified so as to coincide with the case of the perfectly conducting sphere (1). Regarding i2, Eq. [-51 is valid for N I ~< 1.00. For N I in the range 1.00 ~< N I ~< 20.00 Eq. [-51 is not valid since the linear patterns characterizing the points (Fig. 1) become rather erratic. For N I >~ 20.00 the patterns become the same as those for the perfectly conducting sphere (1).

A single calculation with a in the range 4 < a < 24, N R = 1.40, and N I varying from 0.010 to 0.220 was done with the m a x i m a counted from the backward direction. I t was found that for N I _< 0.040 there is a linear pattern, regarding il, i2, and the average of these, for the first and second m a x i m a with intercepts at 1.40 and 3.10, respectively. For N I > 0.040, the patterns for the first and second m a x i m a become erratic and tend to acquire a positive slope. This indicates that the scattering curves become flat in the backward direction as N I becomes large. REFERENCES 1. 1)ATITSAS~

A. J., I E E E Trans. Gen. Appl. 21, 243

(1973).

Journal of Colloid and Interface Science, V o l . 46, N o . 2, F e b r u a r y

1974~

282

PATITSAS

2. PATITSAS,A. J., y. Colloid Interface Sd. 45, 359 (1973). 3. PAX~TSAS,A. J., J. Colloid Interface ScL 46, 261 (1974). 4. DANDLI~R, W. B., J. Amer. Chem. Soc. ?2, 5110 (1950). 5. M~.RON, S. H., AND ELDER, M. E., Y. Colloid Sci. 18, 107 (1963).

6. KEI~E~, M., FARONE, W. A., StaTIC, L. B., AND MATIJEVId, E., Y. Colloid Sci. 19, 193 (1964). 7. KE~I~EI~, M., "The Scattering of Light and other Electromagnetic Radiation." Academic Press, New York, 1969.

Journal of Colloid and Interface Science, VoL 46, No. 2, February 1974