Light scattering by oriented particles

Letter to the Editors Light Scattering by Oriented Particles In the previous paper (1), expressions for depolarization factors for externally oriented dipole scatterers were derived. Here, expressions for the intensity scattered at an angle 0 and the total scattering coeffident are derived. For a cloud of particles oriented under an external field, it can be shown that

p" = ~

'f

Y~E[Jip°J~r] dn i = a,b,c,d,

plane perpendicular to the scattering plane [Ref. (1) for the notations-]. If the particles are assumed to be spheroids with symmetric axis making an angle 3"with the direction 0Y the external field, then the scattering matrix is given by

"~+ (~-~) × cos2,,/cose

i =iR ~

[1]

Substituting Eqs. [63 and [-7-1into the Eqs. [-4] and [-5] and integrating over 3' and q~ gives

FLIoz , 0 ] = I.-Lao L4ot"

[2]

NR*

I~ = ~ /£ is the mean propagation constant and R is the distance of observation. J is the scattering matrix and T denotes its conjugate transpose. Superscripts a, b, c, d denote the four positions of the particles described by Van de Hulst (1, 3a), and dn is the orientation distribution of the particles given by

dn =

N exp ( -- u/KT)

[-16a2a=* + ¢ ( h ) ( 2 X + 4Y) + 3X¢(h)

+ 3X + 4Y -- sin28{8a=~ * + 2¢(h) (X + 4Y) + 7X¢(h) -- X}~[o,,, NR~ L = T~ T E ~ 6 ~ * + 8Y + 6 x - 4¢(h)(x + 2Y)

-- 5X¢(h)}3Io~,

[3]

[9-1

where

x = (,~ - ~2) (~1" - ~*),

where U is the potential energy of a particle in the external field, K is the Boltzmann constant, T is the absolute temperature, and N is the number of particles. U depend mainly on the magnetic properties of the particles. This function for the three different cases was evaluated in the previous paper (1). Scattered intensity is given by the trace relation

Y = ala2* -- 2a2a2* + a2al*. (h) and ~b(h) are the functions appropriate to the three cases considered earlier (1). Thus for spheroids oriented parallel to the field ,(h) = ~

[4]

Substituting Eq. [.1] into Eq. [3] and using the appropriate forms of J i we obtain

(h) = ~

l (hexph' ~

~

1)

1 (hexph'(h,_~)+~) \

~

where ]$2 =

I~,= 2f (JnJ* n + JmJ*~2+ JmJ*m+ J=lJ*m)dnLlo, [5] Eh = I, = f (2J=~l*=2+J,~l*x2+ J2~J%OdnL4o,

[-83

-- 2X~(h) -- sin~O{6X¢(h) -- X ,

f f fo 2~ exp(--u/Kr) sin~d3"&b

I = T~o'.

-- (a,--a2) sin'7 Xcos3, sin~bsin0 (~1-~) sin3, cos-/cos0

where p" is tile coherency matrix for the scattered light and o° is the coherency matrix (2) for the incident light. It is given by

po

(al--a~) sin-/cos-/1 ×cos¢ cos0 | -- (m--a~) sin~3,[ [7] × cos~ sine sin0| (m-a2) sin~3, [ Xcos~¢+~ J

VII 2 2KT

( X l - - X2),

expx2dx.

[.6] V is the volume of the particle, H is the applied magnetic field, and xi is the magnetic susceptibility along the three axes.

where the subscripts u and r denote that the incident light is, respectively, unpolarized and polarized in a 286

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

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

287

L E T T E R TO T H E EDITORS Details of the other two cases can be referred to from

(1). The total scattering coefficient is obtained by integration of Iu and I , over the surface of sphere with radius R 4~R 4 (Csc,)~ = ~ C6¢(h) (X - 210 - 15X~(h) + 36Y + 33X + 96~x2"/8],

1-10"]

4~K 4

(C,o~), = - ~ - E24~a~* + 12V + 10X --12¢ (~) (X + 10 +2X~(h)3.

Cl1~

For random orientation the expression for I~ and (Cs,a)u reduces to that given by Van de Hulst (3b).

REFERENCES 1. MEtITA, R. V., SHAH, I-I. S., AND DESAI, J. N., J. Colloid Interface Sci. 36, 80 (1971). 2. BORN, N., AlVO WOLF, E., "Principles of Optics," pp. 544. Pergamon, Elmsford; N. Y., 1970. 3. VAN DE HULST, H. C., "Light Scattering by Small Particles," Wiley, New York, 1957, (a) p. 49; (b) p. 82. R. V. MEHTA R. V. DESAI

Department of Physics S. V. Regional College of Engineering and Technology Surat, India Received September 19, 1973; accepted October 22, 1973

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