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Scattering by ellipsoids of revolution a comparison of theoretical methods
Scattering by Ellipsoids of Revolution A Comparison of Theoretical Methods PAUL LATIMER Physics Department, Auburn University, Auburn, Alabama 36830 AND
PETER BARBER Department o f Bioengineering, University o f Utah, Salt Lake City, Utah 84112 Received May 2, 1977; accepted May 17, 1977
New methods were recently developed for calculating the differential and total scattering cross sections of ellipsoids of revolution. In one, the extended boundary condition method numerically solves Maxwell's equations exactly for this and other particle shapes. In the other, postulates of the Rayleigh-Gans-Debye and anomalous diffraction approximations and the Lorenz-Mie equations are used to obtain improved approximate methods for ellipsoids. In this paper, these methods are used to calculate differential and total scattering cross sections and the corresponding scattering efficiencies of prolate and oblate ellipsoids. Comparisons of the results provide an independent check of each method. They also supply a basis for assessing the roles of the various mechanisms of the wave-particle interaction.
INTRODUCTION
and other particle shapes (2). Less general methods had been offered by Asano and Yamamoto (3) and by Greenberg and coworkers (4, 5). Each of these new methods is known to approach the proper limit as the axial ratio approaches one, i.e., when the ellipsoid becomes a sphere. However, there are obvious advantages of an independent check of the methods for other shapes. Presently, for four sets of particle parameters we use the new exact (2) and approximate (1) methods to calculate the differential scattering cross sections and total scattering cross section and compare the results. A new version of one of the approximate methods is also proposed here and its predictions are compared with the others.
Of practical interest are the light-scattering characteristics of many particles for which the well-known closed-form solutions of Maxwell's equations are not available, i.e., particles other than homogeneous and coated spheres or infinite cylinders. One model that can approximate most particle shapes is the ellipsoid of revolution. Two new methods were recently developed for predicting the scattering cross sections of such ellipsoids. One uses the RayleighGans-Debye or anomalous diffraction approximation to define an equivalent sphere and the exact Lorenz-Mie relations to determine the scattering by the equivalent sphere (1). In the other, the extended boundary condition method (EBCM) solves Maxwell's equations numerically for this 310 0021-9797/78/0632-0310502.00/0 Copyright© 1978by AcademicPress,Inc. All rightsof reproductionin any formreserved.
Journal of Colloidand Interface Science, Vol.63, No. 2, February 1978
311
S C A T T E R I N G BY E L L I P S O I D S
:t Y
Io
)
"X
(a)
(b)
FIG. 1. Schematic diagrams. Part (a) s h o w s the modified laboratory coordinate s y s t e m , the incident (I0) and scattered b e a m s (Is), the particle axis (heavy arrow), and relevant angles. Part (b) s h o w s a n o t h e r laboratory coordinate s y s t e m , x* - y* - z, u s e d in one approximate method.
TERMINOLOGY
The ellipsoid is defined in a particle coordinate system by ( x / a v ) 2 + (y/a) 2 + (z/a) z = 1,
= e -NLRsc,
R~c =
(r sin OdrbdO,
[3]
[1]
where v is the axial ratio and a is the length of the y- and z-semi-axes. In the original laboratory coordinate system, the incident beam is in the +y direction. Scattering is at the polar angle 0 and the azmuthal angle qb. For computational purposes, simplification is achieved by relating the particle to a modified laboratory coordinate system obtained by rotating the original one by an angle qb about the y-axis. Then the photocell always lies in the x - y plane as shown in Fig. la. In this case, the particle axis makes a polar angle • with the y-axis of this modified laboratory coordinate system and an angle Y as measured from the x-axis in the x - z plane. The total scattering cross section of the particle, Rsc, can be defined in terms of the exponential transmittance equation for a well-collimated beam and small photocell: T
to the differential scattering cross section (6, 7), (r, by
[2]
where N is the number of particles per unit volume, and L is the path length of the beam in the particle system. R~c, a measure of total scattering at all angles, is related
where o- is a function of both q~ and 0. o- can be resolved into components (6) corresponding to vertical and horizontal polarizations which are respectively perpendicular and parallel to the horizontal plane containing the incident and scattered beams. For unpolarized incident light, o- = (O-v + o-u)/2. It is related to o'D used earlier (2) by o- = o-D/41r. Instead of Rsc and (r-values, which require specifying a particular particle size and wavelength, we will use the normalized quantity G = 4"rro'/A, the angular gain (8), and Ksc = R s J A . With a view to the aspherical particle, we define this A as the projected area presented to the beam by the particle. THE APPROXIMATE METHOD
The exact extended boundary condition method is compared below with various forms of the approximate method. Two forms of the latter are based in part on the R a y l e i g h - G a n s - D e b y e approximation and one on the anomalous diffraction approximation. Each form uses the approxiJournal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
312
LATIMER
AND BARBER
TABLE
I
A p p r o x i m a t e M e t h o d s f o r C a l c u l a t i n g ~re o f t h e E l l i p s o i d o f E q . [1] o f R e f r a c t i v e I n d e x me O r i e n t e d w i t h A x i s o f S y m m e t r y M a k i n g a P o l a r A n g l e 't r w i t h t h e B e a m , a n A z m u t h a l A n g l e y a
Method
Radius and refractive index of equivalent sphere
1. R M - I
a c , me
F
Definitions c = (sin 2 8 + v z c o s 2 6)
v2/c 6
2. R M - I I
a c , m~
1
m s = 1 + (me -
3. A M - I
a g ' , m~
g2/(g,)4
g = ( c o s 2 ~ + v ~ sin 2 ~),t~ g ' = (sin2 Y + g~ c o s 2 Y ) 1/2 ms = 1 + ( m e -
4. A M - I I
ag', m s
g2/(g,)4
g = (cos ~~.
1)v/c ~
1)v/(g' g)
+ v 2 sin 2 ~2)1/2
g, = (sinSy , + g2 cos2Y,),/2 ms = 1 + (me - 1 ) v / ( g ' g)
a The equivalent sphere is of radius a c or ag' and refractive index me or ms. For clarity, some of the symbols have been changed from those used in Table I of Ref. (1). In that reference, g' was defined correctly in Table I, but incorrectly in Eq. [5b].
mation to define the parameters o f an equivalent sphere. The L o r e n z - M i e relations are then used to calculate o-, or Rse, of that sphere. The result is in turn multiplied by a normalization factor from the approximation: i.e., or e =
o'sF.
[4]
These existing methods are designated by RM-I, RM-II, and AM-I. During this study it became apparent that the AM-I fails badly at large angles for the particles considered here. H e n c e , we sought another method in better agreement with the EBCM. In the anomalous diffraction approximation (9, 10) the amplitude of the scattered wave is proportional to the projected area of the particle. In the case of the sphere for which van de Hulst developed the method, this is the area seen by the beam. An equal area would also be seen from the photocell. H o w e v e r , these two areas would be different for an ellipsoid revolution. By trial, we found that predictions in better agreement with the E B C M could be obtained when one uses the area seen along Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
a line which bisects 0, the y*-axis in Fig. lb. In AM-I, the sphere parameters and correction factors are defined in terms of the angles • and Y. In AM-II, they are similarly defined in terms o f ~ * and Y*. • * is the angle between the axis of symmetry of the ellipsoid and y* in Fig. lb in the scattering plane. Y* is the azimuthal angle in the x * - z plane which is measured from x*. These approximate methods for obtaining o-e are summarized in Table I. COMPARISON OF THEORETICAL PREDICTIONS
The angular gains, Gv = 47rO-v/A and of four ellipsoids in a given orientation were calculated with the EBCM, RM-I, RM-II, AM-I, and AM-II at 2 ° intervals for 0 = 0 - 1 8 0 °. In this case, A = 7ra2(cos 2 ~ + v 2 sin 2 ~),/2. The particle models are combinations of two shapes and two refractive indices. For each, incidence is assumed to be parallel to the axis o f symmetry ( ~ = 0°). Figure 2 shows Gv (left) and Gh (right) of the soft nonabsorbing prolate ellipsoid (n = 1.05). A scale G h = 47ro-n/A,
SCATTERING BY ELLIPSOIDS
101
00"
M
~
-
10-1
~
'==~.10- -2
_~r~
oEBCM
,
313
~--~-~
RM-II
, AM-I
~f~
o
10-80
I
60
I
120
-180 0
6 (degrees)
I
60
120
--180
0 (degrees)
FIG. 2. The angular gains, Gv (left) and Gh (right), of a prolate ellipsoid of revolution for end-on incidence, where k a = 2 r c a / h = 2.37, v = 3.0, and n = 1.05. All curves connect points at 2° intervals. The absolute values of ordinates are significant; no normalization factors were used. The sketch of wavelength and particle dimensions is scale. sketch of the particle and the incident w a v e is a l s o g i v e n . E x c e p t for A M - I , all o f t h e G - c u r v e s a r e in g o o d a g r e e m e n t , e s p e c i a l l y at s m a l l a n g l e s . T h e a b s o l u t e v a l u e o f e a c h G is significant: no n o r m a l i z a t i o n f a c t o r s w e r e u s e d in p l o t t i n g . F i g u r e 3 s h o w s c o r r e s p o n d i n g curves for a harder prolate ellipsoid (n = 1.20). T h e a g r e e m e n t o f t h e a p p r o x i m a t e m e t h o d s w i t h t h e e x a c t o n e is n o t
TABLE II Ksc, the Total Scattering Efficiency of the Ellipsoids of Figs. 2-5 Obtained with Eq. [3] and Coordinates of the Curves in Those Figures ka Method
EBCM RM-I RM-II AM-I AM-II
= 2.37
ka
= 2.37
ka
= 4.309
ka
= 4.309
v = 3.0
v = 3.0
v = 0.5
v = 0.5
n = 1.05
n = 1.20
n = 1.05
n = 1.20
0.121 0.110 0.115 0.207 0.116
2.65 1.92 1.87 3.10 1.84
0.0226 0.0231 0.0233 0.0211 0.0228
0.387 0.388 0.415 0.351 0.415
q u i t e as g o o d . S u c h a r e d u c t i o n in a c c u r a c y was expected; both underlying approximations (Rayleigh-Gans-Debye and anoma l o u s d i f f r a c t i o n ) a r e i n t e n d e d f o r In - 1 I 1. O t h e r c a l c u l a t i o n s , f o r n -> 1.50, indicate that such particles are outside the domain of the present approximate m e t h o d s . I n Figs. 3 - 5 , t h e c u r v e s f r o m AM-I have been truncated outside their useful domain. F i g u r e 4 s h o w s t h e o r e t i c a l G - c u r v e s for a soft o b l a t e e l l i p s o i d . A g a i n t h e a g r e e m e n t is g o o d . F i n a l l y , Fig. 5 s h o w s the curves for a harder oblate ellipsoid of the s a m e size. I n this c a s e , t h e e f f e c t s o f t h e l a r g e r r e f r a c t i v e i n d e x a r e less n o t i c e a b l e t h a n for t h e p r o l a t e e l l i p s o i d . Of the various approximations, the one w h i c h is f o u n d to a g r e e b e s t w i t h t h e E B C M is R M - I . H o w e v e r , n o t e t h a t t h e s e particles are only somewhat larger than h (see d i a g r a m s ) . F o r l a r g e r p a r t i c l e s , A M - I I w o u l d b e e x p e c t e d to b e m o r e a c c u r a t e a n d R M - I less a c c u r a t e . Journal o f Colloid and Interface Science,
Vo]. 63, No.
2, February
1978
314
LATIMER AND BARBER
10 =
10 ~
:
10 0
=
RM-I
~
RM-II
,
, AM-I
~
=
AM-II
161 10 2
103
10-40
6~0 0
120
180 0
(degrees)
60
120
180
# (degrees)
FIG. 3. The angular gains, Gv and Gh, of a prolate ellipsoid: ka = 2.37, v = 3.0, and t7 = 1.20.
The total scattering cross section Rsc of a particle can be calculated by each method using Eq. [3] which integrates over all angles. The o--values of these figures were used to obtain R~ and in turn K~ by each method. The results for each ellipsoid are shown in Table II. They are as expected; all approximate methods except AM-I substantially agree with the EBCM. K~ can also be evaluated directly in both the EBCM and Lorenz-Mie formulations from the forward amplitude. The previous such approximate methods (1) will be designated as ADLM-I and ADLM-II. These two approximations and the EBCM were used to obtain K~c for the above ellipsoids. The results are given in Table III. The approximate methods are found to be reasonably accurate for all except the long thin and soft ellipsoid, end-on incidence. CONCLUSIONS
The exact EBCM and the present approximate methods for predicting o- and Ksc Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
are found to substantially agree in most cases studied. The residual differences seem to originate with simplifying assumptions of the approximations. The convenience and flexibility of the approximations make them advantageous for some problems (11, 12). However, for other cases, only the exact EBCM method would suffice. The present results support
T A B L E III Ksc, the Total Scattering Efficiency of Ellipsoids of Figs. 2 - 5 Evaluated Directly with the E x t e n d e d B o u n d a r y Condition Method of Ref. (2), from Two Methods, A D L M - I and A D L M - I I o f Ref. (1), and by van de H u l s t ' s Original A n o m a l o u s Diffraction Method for a Sphere o f Equal T h i c k n e s s Particle Method
1
2
3
4
EBCM ADLM-I ADLM-II
0.121 0.246 0.207 0.246
2.65 2.93 3.09 2.23
0.0225 0.0168 0.0210 0.0231
0.385 0.295 0.351 0.356
vdH
SCATTERING BY ELLIPSOIDS
315
lO°I o
o EBCM [] R M - I
101 =
~ RM-II
,
~--
AM-I
,
,, A M - I I
,,
10=
'D
g
104
10 4
11~5
0
I
I
I
60
120
180 0
60
120
180
6 (degrees)
# (degrees)
FIG. 4. The angular gains, Gv and Gh, of an oblate ellipsoid: ka = 4.31, v = 0.5, and n = 1.05.
the tested methods,
furnish information as
to the domains of the approximations,
and
provide
a better
the wave-particle
basis for understanding interaction.
10 ° o
o EBCM
=
= RM-I
l(J' .
~,= 10
-
RM-II
•
AM-I
2
10-3
lO-,I 1(~5I 0
60 0
120 (degrees)
180 0
60 #
120
180
(degrees)
FIo. 5. The angular gains, Gv and Gh, of an oblate ellipsoid: ka = 4.31, v = 0.5, and n = 1.20. Journal of Colloid and Interface Science, Vol. 63, No. 2, F e b r u a r y 1978
316
LATIMER AND BARBER REFERENCES
1. Latimer, P., J. Colloid Interface Sci. 53, 102 (1975). 2. Barber, P., and Yeh, C., Appl. Opt. 14, 2864 (1975). 3. Asano, S., and Yamamoto, G., Appl. Opt. 14, 29 (1975). 4. Greenberg, J. M., Lind, A. C., Wang, R. T., and Libelo, L. F., in "Electromagnetic Scattering (ICES)" (M. Kerker, Ed.), p. 123. Pergamon, London, 1963. 5. Greenberg, J. M., Lind, A. C., and Wang, R. T., in "Electromagnetic Scattering (ICES II)" (R. L. Rowell and R. S. Stein, Eds.), p. 3. Gordon & Breach, New York, 1967.
Journal of CoUoid and Interface Science, Vol. 63, No. 2, February 1978
6. Deirmendjian, D., "Electromagnetic Scattering by Spherical Polydispersions," p. 13. Amer. Elsevier, New York, 1969. 7. Schiff, L. I., "Quantum Mechanics," 1st ed., p. 106. McGraw-Hill, New York, 1949. 8. Kerker, M., "The Scattering of Light and Other Electromagnetic Radiations," p. 156. Academic Press, New York, 1969. 9. Van de Hulst, H. C., "Light Scattering by Small Particles," p. 172. Wiley, New York, 1949. 10. Moore, D. M., Bryant, F. D., and Latimer, P., J. Opt. Soc. Amer. 58, 281 (1968). 11. Latimer, P., Appl. Opt. 14, 2324 (1975). 12. Latimer, P., Born, G. V. R., and Michel, F., Arch. Biochem. Biophys. 180, 151 (1977).
PETER BARBER Department o f Bioengineering, University o f Utah, Salt Lake City, Utah 84112 Received May 2, 1977; accepted May 17, 1977
New methods were recently developed for calculating the differential and total scattering cross sections of ellipsoids of revolution. In one, the extended boundary condition method numerically solves Maxwell's equations exactly for this and other particle shapes. In the other, postulates of the Rayleigh-Gans-Debye and anomalous diffraction approximations and the Lorenz-Mie equations are used to obtain improved approximate methods for ellipsoids. In this paper, these methods are used to calculate differential and total scattering cross sections and the corresponding scattering efficiencies of prolate and oblate ellipsoids. Comparisons of the results provide an independent check of each method. They also supply a basis for assessing the roles of the various mechanisms of the wave-particle interaction.
INTRODUCTION
and other particle shapes (2). Less general methods had been offered by Asano and Yamamoto (3) and by Greenberg and coworkers (4, 5). Each of these new methods is known to approach the proper limit as the axial ratio approaches one, i.e., when the ellipsoid becomes a sphere. However, there are obvious advantages of an independent check of the methods for other shapes. Presently, for four sets of particle parameters we use the new exact (2) and approximate (1) methods to calculate the differential scattering cross sections and total scattering cross section and compare the results. A new version of one of the approximate methods is also proposed here and its predictions are compared with the others.
Of practical interest are the light-scattering characteristics of many particles for which the well-known closed-form solutions of Maxwell's equations are not available, i.e., particles other than homogeneous and coated spheres or infinite cylinders. One model that can approximate most particle shapes is the ellipsoid of revolution. Two new methods were recently developed for predicting the scattering cross sections of such ellipsoids. One uses the RayleighGans-Debye or anomalous diffraction approximation to define an equivalent sphere and the exact Lorenz-Mie relations to determine the scattering by the equivalent sphere (1). In the other, the extended boundary condition method (EBCM) solves Maxwell's equations numerically for this 310 0021-9797/78/0632-0310502.00/0 Copyright© 1978by AcademicPress,Inc. All rightsof reproductionin any formreserved.
Journal of Colloidand Interface Science, Vol.63, No. 2, February 1978
311
S C A T T E R I N G BY E L L I P S O I D S
:t Y
Io
)
"X
(a)
(b)
FIG. 1. Schematic diagrams. Part (a) s h o w s the modified laboratory coordinate s y s t e m , the incident (I0) and scattered b e a m s (Is), the particle axis (heavy arrow), and relevant angles. Part (b) s h o w s a n o t h e r laboratory coordinate s y s t e m , x* - y* - z, u s e d in one approximate method.
TERMINOLOGY
The ellipsoid is defined in a particle coordinate system by ( x / a v ) 2 + (y/a) 2 + (z/a) z = 1,
= e -NLRsc,
R~c =
(r sin OdrbdO,
[3]
[1]
where v is the axial ratio and a is the length of the y- and z-semi-axes. In the original laboratory coordinate system, the incident beam is in the +y direction. Scattering is at the polar angle 0 and the azmuthal angle qb. For computational purposes, simplification is achieved by relating the particle to a modified laboratory coordinate system obtained by rotating the original one by an angle qb about the y-axis. Then the photocell always lies in the x - y plane as shown in Fig. la. In this case, the particle axis makes a polar angle • with the y-axis of this modified laboratory coordinate system and an angle Y as measured from the x-axis in the x - z plane. The total scattering cross section of the particle, Rsc, can be defined in terms of the exponential transmittance equation for a well-collimated beam and small photocell: T
to the differential scattering cross section (6, 7), (r, by
[2]
where N is the number of particles per unit volume, and L is the path length of the beam in the particle system. R~c, a measure of total scattering at all angles, is related
where o- is a function of both q~ and 0. o- can be resolved into components (6) corresponding to vertical and horizontal polarizations which are respectively perpendicular and parallel to the horizontal plane containing the incident and scattered beams. For unpolarized incident light, o- = (O-v + o-u)/2. It is related to o'D used earlier (2) by o- = o-D/41r. Instead of Rsc and (r-values, which require specifying a particular particle size and wavelength, we will use the normalized quantity G = 4"rro'/A, the angular gain (8), and Ksc = R s J A . With a view to the aspherical particle, we define this A as the projected area presented to the beam by the particle. THE APPROXIMATE METHOD
The exact extended boundary condition method is compared below with various forms of the approximate method. Two forms of the latter are based in part on the R a y l e i g h - G a n s - D e b y e approximation and one on the anomalous diffraction approximation. Each form uses the approxiJournal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
312
LATIMER
AND BARBER
TABLE
I
A p p r o x i m a t e M e t h o d s f o r C a l c u l a t i n g ~re o f t h e E l l i p s o i d o f E q . [1] o f R e f r a c t i v e I n d e x me O r i e n t e d w i t h A x i s o f S y m m e t r y M a k i n g a P o l a r A n g l e 't r w i t h t h e B e a m , a n A z m u t h a l A n g l e y a
Method
Radius and refractive index of equivalent sphere
1. R M - I
a c , me
F
Definitions c = (sin 2 8 + v z c o s 2 6)
v2/c 6
2. R M - I I
a c , m~
1
m s = 1 + (me -
3. A M - I
a g ' , m~
g2/(g,)4
g = ( c o s 2 ~ + v ~ sin 2 ~),t~ g ' = (sin2 Y + g~ c o s 2 Y ) 1/2 ms = 1 + ( m e -
4. A M - I I
ag', m s
g2/(g,)4
g = (cos ~~.
1)v/c ~
1)v/(g' g)
+ v 2 sin 2 ~2)1/2
g, = (sinSy , + g2 cos2Y,),/2 ms = 1 + (me - 1 ) v / ( g ' g)
a The equivalent sphere is of radius a c or ag' and refractive index me or ms. For clarity, some of the symbols have been changed from those used in Table I of Ref. (1). In that reference, g' was defined correctly in Table I, but incorrectly in Eq. [5b].
mation to define the parameters o f an equivalent sphere. The L o r e n z - M i e relations are then used to calculate o-, or Rse, of that sphere. The result is in turn multiplied by a normalization factor from the approximation: i.e., or e =
o'sF.
[4]
These existing methods are designated by RM-I, RM-II, and AM-I. During this study it became apparent that the AM-I fails badly at large angles for the particles considered here. H e n c e , we sought another method in better agreement with the EBCM. In the anomalous diffraction approximation (9, 10) the amplitude of the scattered wave is proportional to the projected area of the particle. In the case of the sphere for which van de Hulst developed the method, this is the area seen by the beam. An equal area would also be seen from the photocell. H o w e v e r , these two areas would be different for an ellipsoid revolution. By trial, we found that predictions in better agreement with the E B C M could be obtained when one uses the area seen along Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
a line which bisects 0, the y*-axis in Fig. lb. In AM-I, the sphere parameters and correction factors are defined in terms of the angles • and Y. In AM-II, they are similarly defined in terms o f ~ * and Y*. • * is the angle between the axis of symmetry of the ellipsoid and y* in Fig. lb in the scattering plane. Y* is the azimuthal angle in the x * - z plane which is measured from x*. These approximate methods for obtaining o-e are summarized in Table I. COMPARISON OF THEORETICAL PREDICTIONS
The angular gains, Gv = 47rO-v/A and of four ellipsoids in a given orientation were calculated with the EBCM, RM-I, RM-II, AM-I, and AM-II at 2 ° intervals for 0 = 0 - 1 8 0 °. In this case, A = 7ra2(cos 2 ~ + v 2 sin 2 ~),/2. The particle models are combinations of two shapes and two refractive indices. For each, incidence is assumed to be parallel to the axis o f symmetry ( ~ = 0°). Figure 2 shows Gv (left) and Gh (right) of the soft nonabsorbing prolate ellipsoid (n = 1.05). A scale G h = 47ro-n/A,
SCATTERING BY ELLIPSOIDS
101
00"
M
~
-
10-1
~
'==~.10- -2
_~r~
oEBCM
,
313
~--~-~
RM-II
, AM-I
~f~
o
10-80
I
60
I
120
-180 0
6 (degrees)
I
60
120
--180
0 (degrees)
FIG. 2. The angular gains, Gv (left) and Gh (right), of a prolate ellipsoid of revolution for end-on incidence, where k a = 2 r c a / h = 2.37, v = 3.0, and n = 1.05. All curves connect points at 2° intervals. The absolute values of ordinates are significant; no normalization factors were used. The sketch of wavelength and particle dimensions is scale. sketch of the particle and the incident w a v e is a l s o g i v e n . E x c e p t for A M - I , all o f t h e G - c u r v e s a r e in g o o d a g r e e m e n t , e s p e c i a l l y at s m a l l a n g l e s . T h e a b s o l u t e v a l u e o f e a c h G is significant: no n o r m a l i z a t i o n f a c t o r s w e r e u s e d in p l o t t i n g . F i g u r e 3 s h o w s c o r r e s p o n d i n g curves for a harder prolate ellipsoid (n = 1.20). T h e a g r e e m e n t o f t h e a p p r o x i m a t e m e t h o d s w i t h t h e e x a c t o n e is n o t
TABLE II Ksc, the Total Scattering Efficiency of the Ellipsoids of Figs. 2-5 Obtained with Eq. [3] and Coordinates of the Curves in Those Figures ka Method
EBCM RM-I RM-II AM-I AM-II
= 2.37
ka
= 2.37
ka
= 4.309
ka
= 4.309
v = 3.0
v = 3.0
v = 0.5
v = 0.5
n = 1.05
n = 1.20
n = 1.05
n = 1.20
0.121 0.110 0.115 0.207 0.116
2.65 1.92 1.87 3.10 1.84
0.0226 0.0231 0.0233 0.0211 0.0228
0.387 0.388 0.415 0.351 0.415
q u i t e as g o o d . S u c h a r e d u c t i o n in a c c u r a c y was expected; both underlying approximations (Rayleigh-Gans-Debye and anoma l o u s d i f f r a c t i o n ) a r e i n t e n d e d f o r In - 1 I 1. O t h e r c a l c u l a t i o n s , f o r n -> 1.50, indicate that such particles are outside the domain of the present approximate m e t h o d s . I n Figs. 3 - 5 , t h e c u r v e s f r o m AM-I have been truncated outside their useful domain. F i g u r e 4 s h o w s t h e o r e t i c a l G - c u r v e s for a soft o b l a t e e l l i p s o i d . A g a i n t h e a g r e e m e n t is g o o d . F i n a l l y , Fig. 5 s h o w s the curves for a harder oblate ellipsoid of the s a m e size. I n this c a s e , t h e e f f e c t s o f t h e l a r g e r r e f r a c t i v e i n d e x a r e less n o t i c e a b l e t h a n for t h e p r o l a t e e l l i p s o i d . Of the various approximations, the one w h i c h is f o u n d to a g r e e b e s t w i t h t h e E B C M is R M - I . H o w e v e r , n o t e t h a t t h e s e particles are only somewhat larger than h (see d i a g r a m s ) . F o r l a r g e r p a r t i c l e s , A M - I I w o u l d b e e x p e c t e d to b e m o r e a c c u r a t e a n d R M - I less a c c u r a t e . Journal o f Colloid and Interface Science,
Vo]. 63, No.
2, February
1978
314
LATIMER AND BARBER
10 =
10 ~
:
10 0
=
RM-I
~
RM-II
,
, AM-I
~
=
AM-II
161 10 2
103
10-40
6~0 0
120
180 0
(degrees)
60
120
180
# (degrees)
FIG. 3. The angular gains, Gv and Gh, of a prolate ellipsoid: ka = 2.37, v = 3.0, and t7 = 1.20.
The total scattering cross section Rsc of a particle can be calculated by each method using Eq. [3] which integrates over all angles. The o--values of these figures were used to obtain R~ and in turn K~ by each method. The results for each ellipsoid are shown in Table II. They are as expected; all approximate methods except AM-I substantially agree with the EBCM. K~ can also be evaluated directly in both the EBCM and Lorenz-Mie formulations from the forward amplitude. The previous such approximate methods (1) will be designated as ADLM-I and ADLM-II. These two approximations and the EBCM were used to obtain K~c for the above ellipsoids. The results are given in Table III. The approximate methods are found to be reasonably accurate for all except the long thin and soft ellipsoid, end-on incidence. CONCLUSIONS
The exact EBCM and the present approximate methods for predicting o- and Ksc Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
are found to substantially agree in most cases studied. The residual differences seem to originate with simplifying assumptions of the approximations. The convenience and flexibility of the approximations make them advantageous for some problems (11, 12). However, for other cases, only the exact EBCM method would suffice. The present results support
T A B L E III Ksc, the Total Scattering Efficiency of Ellipsoids of Figs. 2 - 5 Evaluated Directly with the E x t e n d e d B o u n d a r y Condition Method of Ref. (2), from Two Methods, A D L M - I and A D L M - I I o f Ref. (1), and by van de H u l s t ' s Original A n o m a l o u s Diffraction Method for a Sphere o f Equal T h i c k n e s s Particle Method
1
2
3
4
EBCM ADLM-I ADLM-II
0.121 0.246 0.207 0.246
2.65 2.93 3.09 2.23
0.0225 0.0168 0.0210 0.0231
0.385 0.295 0.351 0.356
vdH
SCATTERING BY ELLIPSOIDS
315
lO°I o
o EBCM [] R M - I
101 =
~ RM-II
,
~--
AM-I
,
,, A M - I I
,,
10=
'D
g
104
10 4
11~5
0
I
I
I
60
120
180 0
60
120
180
6 (degrees)
# (degrees)
FIG. 4. The angular gains, Gv and Gh, of an oblate ellipsoid: ka = 4.31, v = 0.5, and n = 1.05.
the tested methods,
furnish information as
to the domains of the approximations,
and
provide
a better
the wave-particle
basis for understanding interaction.
10 ° o
o EBCM
=
= RM-I
l(J' .
~,= 10
-
RM-II
•
AM-I
2
10-3
lO-,I 1(~5I 0
60 0
120 (degrees)
180 0
60 #
120
180
(degrees)
FIo. 5. The angular gains, Gv and Gh, of an oblate ellipsoid: ka = 4.31, v = 0.5, and n = 1.20. Journal of Colloid and Interface Science, Vol. 63, No. 2, F e b r u a r y 1978
316
LATIMER AND BARBER REFERENCES
1. Latimer, P., J. Colloid Interface Sci. 53, 102 (1975). 2. Barber, P., and Yeh, C., Appl. Opt. 14, 2864 (1975). 3. Asano, S., and Yamamoto, G., Appl. Opt. 14, 29 (1975). 4. Greenberg, J. M., Lind, A. C., Wang, R. T., and Libelo, L. F., in "Electromagnetic Scattering (ICES)" (M. Kerker, Ed.), p. 123. Pergamon, London, 1963. 5. Greenberg, J. M., Lind, A. C., and Wang, R. T., in "Electromagnetic Scattering (ICES II)" (R. L. Rowell and R. S. Stein, Eds.), p. 3. Gordon & Breach, New York, 1967.
Journal of CoUoid and Interface Science, Vol. 63, No. 2, February 1978
6. Deirmendjian, D., "Electromagnetic Scattering by Spherical Polydispersions," p. 13. Amer. Elsevier, New York, 1969. 7. Schiff, L. I., "Quantum Mechanics," 1st ed., p. 106. McGraw-Hill, New York, 1949. 8. Kerker, M., "The Scattering of Light and Other Electromagnetic Radiations," p. 156. Academic Press, New York, 1969. 9. Van de Hulst, H. C., "Light Scattering by Small Particles," p. 172. Wiley, New York, 1949. 10. Moore, D. M., Bryant, F. D., and Latimer, P., J. Opt. Soc. Amer. 58, 281 (1968). 11. Latimer, P., Appl. Opt. 14, 2324 (1975). 12. Latimer, P., Born, G. V. R., and Michel, F., Arch. Biochem. Biophys. 180, 151 (1977).