Multiple Light Scattering in Laser Particle Sizing

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Multiple light scattering in laser particle sizing

Alexander A. Kokhanovsky and Reiner Weichert

Multiple light scattering is an important issue in modern laser diffraction spectrometry. Most laserparticle sizers do not account for multiple light scattering in a disperse medium under investigation.This causes an underestimation of the particle sizes in the case of high concentrations of scatterers. Theretrieval accuracy is improved if the measured data are processed with multiple-scattering algorithmsthat treat multiple light scattering in a disperse medium. We evaluate the influence of multiple lightscattering on light transmitted by scattering layers. The relationships among different theories toaccount for multiple light scattering in laser particle sizing are considered. © 2001 Optical Society ofAmerica

OCIS codes: 170.7050, 290.0290, 350.4990.

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1. Introduction

Small-angle laser diffraction is often used for deter-mining particle size distributions ~PSD’s! in dispersemedia.1 The retrieval schemes in this case arebased on the strong dependence of Fraunhofer dif-fraction patterns of large particles on their sizes. In-deed it follows for the scattered light intensity in thecase of monodisperse ensembles of large spheres2

that

I~u, a! 5 NJ1

2~au!a2

u2R2 I0 V, (1)

where J1 is the Bessel function; a is the radius ofparticles; a 5 ka, where k 5 2pyl, where l is thewavelength of incident light; u is the scattering angle,I0 is the intensity of the incident light, N is the num-er of particles per unit volume, V is the volume of acattering medium, and R is the distance to the ob-ervation point. The intensity I~u, a! is proportionalo the square of the Bessel function and therefore haseveral minima and maxima. The positions of min-ma and maxima depend on size parameter a. The

When this research was performed, the authors were with theInstitute of Particle Technology and Environmental Engineering,Technical University Clausthal, Leibnizstrasse 19, D-38678Clausthal-Zellerfeld, Germany. A. A. Kokhanovsky [email protected]! has returned to the Stepanov Institute of Physics, Na-ional Academy of Sciences of Belarus, 70 Skarina Avenue, 220 072insk, Belarus.Received 22 October 1999; revised manuscript received 14 Au-

ust 2000.0003-6935y01y091507-07$15.00y0© 2001 Optical Society of America

rst minimum occurs at scattering angle u 5 Aya,here A ' 3.832.We can account for the polydispersity of a disperseedium by averaging the scattered light intensity

see Eq. ~1!# with the PSD q0~a!:

I#~u, a! 5 *0

`

I~u, a!q0~a!da. (2)

Equation ~1! is valid under the following assump-tions:

a .. 1, 2aum 2 1u .. 1, (3)

lL

.. 1, (4)

Dd

.. 1, (5)

here m 5 n 2 ik is the refractive index of thearticles, L is the geometric thickness of a scatteringayer, l is the mean photon free path length, d is theiameter of the particles, and D is the distance be-ween particles. Note that, owing to assumption ~3!,he Fraunhofer diffraction pattern of a particle coin-ides with the Fraunhofer diffraction pattern for alack screen with radius a. The task of the retrievallgorithm of the laser diffraction spectrometry is tond the radius ~or radii distributions! of such screens.quations ~4! and ~5! state, respectively, that theultiple scattering3 and correlation effects4,5 are neg-

igible.In many problems, such as those encountered in

article sizing for Diesel injectors and heavy fuel oil

20 March 2001 y Vol. 40, No. 9 y APPLIED OPTICS 1507

.oos

ttns

w

T

m

a

H

E

pp

l

c

1

atomizers, the correlation effects can be neglected ~D. d!, but condition ~4! is often not fulfilled. Thusne needs to account for multiple light scattering andbtain a more general formula than the simple single-cattering approximation presented by Eq. ~1!.This could be done by different means. Interest-

ing approaches to dealing with the problem havebeen proposed by Felton et al.6 and Hirleman.7,8 Op-tical particle-sizing techniques based on the use ofthe radiative-transfer equation3 have been studied byBelov et al.,9 Vagin and Veretennikov,10 andSchnablegger and Glatter.11

We show that analytical solutions7–11 are identicalat small angles and can be mutually derived from oneanother. The simplified equation for the calculationof small-angle light-scattering patterns of multiplelight-scattering media with large spherical particlesis derived.

2. Multiple Light Scattering

A. Different Forms of Solutions

The radiative-transfer equation ~RTE! for a plane-parallel light-scattering layer can be written in thefollowing form3:

cos qdI~t, q!

dt5 2I~t, q!

1v0

2 *0

p

I~t, q9!p~u!sin q9dq9, (6)

where q is the observation angle; v0 5 sscaysext is thesingle-scattering albedo, where ssca is the scatteringcoefficient and sext is the extinction coefficient; p~u! ishe azimuthally averaged phase function; t 5 sextL ishe optical thickness, where L is the geometric thick-ess of a layer; and I is the light intensity. Thecattering angle u is related to angles q, q9 by the

following simple equation3:

u 5 arccos@cos q cos q9 1 sin q sin q9 cos ~q 2 q9!#

(7)

here q and q9 are azimuths.The specific form of the RTE presented by Eq. ~6! is

valid only at the normal illumination of a light-scattering plane-parallel layer by a plane wave. Thetransmitted diffuse light intensity does not depend onthe azimuth in this case owing to the symmetry of theproblem.

We consider the angular dependence of functionI~t, q! at small values of the observation angle q.

hus it is assumed that

cos qdI~t, q!

dt<

dI~t, q!

dt, (8)

where we accounted for the approximate equality:cos q ' 1, which is valid as q 3 0. This approxi-

ation is called the small-angle approximation3 ofthe radiative-transfer theory. Equation ~6! under

508 APPLIED OPTICS y Vol. 40, No. 9 y 20 March 2001

approximation ~8! can be solved analytically. Thenswer is well known ~see, e.g., Refs. 3 and 12!:

I~t, q! 5I0

2p (j50

`

Jj~t!Pj~cos q!, (9)

where

Jj~t! 5 Sj 112D exp@2~1 2 v0pj!t#, (10)

pj 512 *

0

p

p~u!Pj~cos u!sin udu. (11)

ere Pj ~cos u! is the Legendre polynomial.Our prime interest here is the incoherent intensity

of multiply scattered light. Thus we remove the co-herent component3 ~or direct light!,

I0 exp~2t! d~cos q 2 1! 5I0

2pexp~2t!

3 (j50

` Sj 112DPj~cos q!,

(12)

from Eq. ~9! and obtain

I9~t, q! 5I0

2p (j50

`

Jj9~t!Pj~cos q!, (13)

where

Jj9~t! 5 Sj 112D e2t@exp~v0pjt! 2 1#. (14)

quations ~13! and ~14! were obtained by Hartel12

and used in Ref. 11 for solution of the inverse prob-lem. Note that numerical calculations of coefficientspj @see Eq. ~11!# are complicated in the case of large

articles because the phase function is stronglyeaked in the forward direction ~u 5 0! at a .. l.Let us derive the integral form of Eq. ~13!. It fol-

ows as u 3 0:

Pj~cos u! 3 J0 FuSj 112DG . (15)

Thus we can introduce the following function, whichfollows from Eqs. ~11! and ~15!:

g~s! 512 *

0

`

p~u!J0~su!udu, (16)

where s 5 j 1 1⁄2 and we extend the upper limit of theintegration in Eq. ~11! to infinity. This is possibleowing to a sharp peak in the phase function p~u! @seeEq. ~1!# at scattering angle u ' 0. Also we use theontinuous function g~s! instead of discrete numbers

taS

ttaamcF

w

Et

ik

w

n

wa

pj @see Eq. ~11!#. It follows from Eq. ~13! and theEuler sum formula

(l50

`

f Sl 112D < *

0

`

f ~p!dp

that

I9~t, q! 5I0

2pe2t *

0

`

$exp@tv0 g~s!# 2 1%J0~sq!sds.

(17)

Equation ~17! is more convenient for applicationshan Eq. ~13!. Note that the simple form of Eq. ~1!llows for the analytical integration in Eq. ~16! ~seeubsection 3.A!.Equation ~17! was used in Refs. 9 and 10 for solu-

ion of the inverse problem taking into account mul-iple light scattering. Thus it is obvious thatpproaches to optical particle sizing in Refs. 9, 10,nd 11 are based on asymptotically equivalent for-ulas @see Eqs. ~13! and ~17!#. Note that Eq. ~17!

an be obtained directly from Eq. ~6! by the use ofourier transform techniques.13

Let us now consider the expansion of the exponentin Eq. ~17!:

exp@tv0 g~s!# 5 (n50

` tnv0ngn~s!

n!. (18)

It follows from Eqs. ~17! and ~18! that

I9~t, q! 5I0

2pe2t (

n51

` tnv0n

n!Hn~q!, (19)

here

Hn~q! 5 *0

`

gn~s!J0~sq!sds. (20)

Note that the successive scattering order expansion@Eq. ~19!# coincides14 with Hirleman’s solution @see

q. ~33! in Ref. 7# and provides a new way to calculatehe function Hn~q!.

B. Single-Scattering Approximation

It follows from Eq. ~19! at n 5 1 and small values oft ~the single-scattering approximation! that

I9~t, q! 5I0

2ptv0 *

0

`

g~s!J0~sq!sds, (21)

or using the equality tv0 5 sscaL yields

I9~t, q! 5I0

2psscaLp~q!, (22)

where @see Eq. ~16!#

p~q! 5 *0

`

g~s!J0~sq!sds (23)

s the phase function. On the other hand, it is wellnown2 that the phase function of the Fraunhofer

diffraction process has the following form:

p~q! 54J1

2~qr!

q2 . (24)

Thus for the intensity within the framework of thesingle-scattering approximation one can obtain

I~t, q! 5 p~q!a2NLI0, (25)

where we use the equality ssca 5 2pNa2, which isvalid at a .. l.3

It follows that Eq. ~25! @see also Eq. ~24!# coincidesith Eq. ~1! at L 5 Vy~4R2!. Different multipliers @L

and Vy~4R2!# are due to the different symmetry of theproblem associated with Eqs. ~1! and ~6!. Note that

ormalized intensities i~t, q! 5 I~t, q!yI~t, 0! definedby Eqs. ~1! and ~25! coincide.

Thus one can see that Eqs. ~13!, ~17!, and ~19! arespecial cases of the solution of the RTE @Eq. ~6!#.They have already been used for development of dif-ferent inversion schemes within the framework oflaser diffraction spectrometry.8,9,11

3. Multiple Fraunhofer Diffraction

A. Monodisperse Media

The main task in this section is to study Eq. ~17! inmore detail for the multiply scattered light intensityI~t, q!. This formula can be applied to many kinds ofdisperse media with different types of forwardpeaked phase function, including Rayleigh–Gansphase functions.3

Let us now consider a special case of Eq. ~17! withthe phase function p~u!, defined in Eq. ~24!. It fol-lows for the Fourier–Bessel transform of the phasefunction ~see Fig. 1! that

g~z! 5 52p

@arccos~z! 2 z~1 2 z2!1y2# z # 1

0 z . 1, (26)

here z 5 syh and h 5 2a. The single-scatteringlbedo v0 is equal to 0.5 within the framework of the

Fraunhofer approximation. Thus it follows from Eq.~17! for the intensity of the transmitted diffused lightthat

I9~t, q! 5I0

2pe2t *

0

` HexpFtg~syh!

2 G 2 1JJ0~sq!sds

(27)

or

I9~t, q! 5 Ce2t *0

1 FexpFtg~z!

2 G 2 1GJ0~bz!zdz, (28)

where b 5 2kaq, C 5 @~2a2!yp#I0, and the functiong~z! is presented by Eq. ~26! and Fig. 1. One canobserve the results of calculations of the normalizedintensity i~t, q! 5 I~t, q!yI~t, 0! obtained with Eq. ~28!


fmac~slst

e

I

a

w3otnTt

tgm

2t

t

1

for different values of the optical thickness t as aunction of parameter b in Fig. 2. It follows thatultiple light scattering causes broadening of the

ngular intensity distributions. The same effectould be due to the decrease in particle size @see Eq.1!#. Thus multiple-scattering broadening is respon-ible for the underestimation of the particle size byaser diffraction spectrometers, designed for thincattering layers, if they are used for large values ofhe optical thickness.

Let us define the dimensionless halfwidth param-ter h by

i~b 5 h! 5 0.5. (29)

t can be calculated from Eq. ~28! for different t. Thevalue of h increases with the optical thickness ~seeFig. 3 and Table 1!. Note that Table 1 can be used toestimate the radii of monodisperse particles from

Fig. 1. Fourier–Bessel transform of the Fraunhofer phase func-tion @Eq. ~24!# obtained with the exact solution @Eq. ~26!# and theapproximate formula @Eq. ~34!#.


measurements of optical thickness t and observationngle q0 at which i~q0! 5 0.5:

a 5h~t!

4pq0l. (30)

The value of h in Eq. ~30! should be selected fromTable 1, depending on optical thickness. Also onecan use h 5 3.23614 1 0.0768t 1 0.00937t2, which

as obtained by fitting the data in Table 1 ~see Fig.!. The same method can be applied for estimationf the mode radius of narrow particle-size distribu-ions. However, note that the values in Table 1 ig-ore light rays reflected and transmitted by particles.hey account only for the diffraction component ofhe total scattered-light intensity.

It is useful to estimate the upper limit of opticalhickness t*, where multiple scattering can be ne-lected. It follows from Fig. 2 that the influence ofultiple light scattering is larger for larger values of

Fig. 3. Dependence of half-width h of the angular spectrum of theransmitted light on optical thickness t.

Table 1. Dependence of the Half-Width Parameter h of the Angular

Fig. 2. Dependence of the normalized intensity on parameter b 5kaq obtained with Eq. ~28! at different values of optical thickness5 0.01, 1, 2, 5, 7.

Distribution of Transmitted Light on Optical Thickness t for DisperseMedia with Monodispersed Spheres

t h

0.01 3.230.5 3.281.0 3.321.5 3.382.0 3.432.5 3.493.0 3.553.5 3.624.0 3.694.5 3.775.0 3.855.5 3.946.0 4.036.5 4.14

st

vtdd

tP

2

p~ot

~iupp

~h

parameter b 5 2kaq. One can introduce the follow-ing criteria on the basis of data in Table 1:

1 2h~t*!

h~0.01!# ε, (31)

where the value of ε depends on the tolerable error forthe retrieval scheme. Clearly, the value of h at t 50.01 corresponds to the case of single scattering; e.g.,if one allows for the difference in half-widths ε 5 1%,it follows that t* # 0.5 to fulfill Eq. ~31!. MalvernInstruments advises that the upper limit of the ob-scuration m 5 @1 2 exp~2t!# 3 100% should be 50% toavoid multiple light scattering.15 The optical thick-ness t* is equal to 0.7 in this case ~see Fig. 4! and ε '2% @see inequality ~31! and Table 1#. Bayvel etal.16,17 found that it is possible to obtain reliable re-sults of the inversion even at m 5 90% ~or at t* 5 2.3,ee Fig. 4!. One can find that the value of ε is equalo 7% in this case.

In conclusion, Eq. ~28! @see Eq. ~26! as well# pro-ides a simple and useful tool for studies of the con-ribution of multiple light scattering to Fraunhoferiffraction patterns of scattering media with mono-isperse spheres.

B. Polydispersions

It is important to generalize Eq. ~27! for the case ofpolydispersions. It could be done by averaging opti-cal thickness t and the Fourier–Bessel transform ofhe phase function g~z! @see Eqs. ~1! and ~26!# with theSD q0~a! ~Ref. 20!:

^t& 5 *0

`

t~a!q0~a!da, (32)

^g~s!& 5

*B

`

g~sy2ka!a2q0~a!da

*0

`

a2q0~a!da

, (33)

Fig. 4. Dependence of the optical thickness on obscuration.

where B 5 sy2k and t~a! 5 2pa NL ~Ref. 3! for largearticles. One should substitute the values of Eqs.32! and ~33! @instead of t and g~syh!# in Eq. ~27! tobtain the expression for the intensity of the lightransmitted through a polydisperse medium.

Thus it follows that the only complication in Eq.27! for a polydispersed medium is that function g~s!s not described by the simple Eq. ~26!. One shouldse the ratio of integrals @Eq. ~33!# in this more com-lex case @see Eq. ~32!#. The optical thickness of aolydisperse medium is determined by the integral *0

`

a2q0~a!da. The integrals in Eqs. ~32! and ~33! can beevaluated analytically for some special cases.

For example, it could be done by expansion of thefunction g~z! @see Eq. ~26!# at small values of z:

g~z! 5 1 1 (j50

3

sjz2j11, (34)

where s0 5 2~4yp!, s1 5 2y~3p!, s2 5 1y~10p!, s3 51y~28p!. Here we neglected terms of the order of z9

and higher. The error in the replacement of Eq. ~26!by Eq. ~34! is small ~see Fig. 1!. The largest error islocated at z ' 1. However, the value of g~z! is smallat small values of 1 2 z@g~1! 5 0# @see Eq. ~26!#, andthe correspondent contribution to Eq. ~27! can be ne-glected.

It follows from Eqs. ~33! and ~34! that

^g~s!& 5 1 1 (j50

3

sjw~2j 1 1, B!, (35)

where

w~n, B! 5 Bn*

B

`

a22nq0~a!da

*0

`

a2q0~a!da

. (36)

The special function w~n, B! in Eq. ~36! can be calcu-lated analytically for many types of PSD. For exam-ple, it follows for the gamma PSD q0~a! 5 Aam

exp~2amya0!, where A is the normalization constant*0

` q0~a!da 5 1!, a0 is the mode radius, and m is thealf-width parameter:

^g~s!& 5 1 1 (j50

3

sjw~2j 1 1, b!,

w~n, b! 5 bn*

b

`

y21m2n exp~2y!dy

*0

`

y21m exp~2y!dy

, (37)

where b 5 ~lms!y~4pa0!. From Eq. ~37! at m . n 23 one can obtain

w~n, b! 5 bn G~m 1 3 2 n!@1 2 P~m 1 3 2 n, b!#

G~m 1 3!,

(38)


a

w

ca

i

0ta

tlmnfdM

asefdd

t

0ac

1

where

G~a! 5 *0

`

ta21 exp~2t!dt (39)

is the gamma function, and

P~a, x! 51

G~a! *0

x

ta21 exp~2t!dt (40)

is the incomplete gamma function. It follows at in-teger values of m that

G~m! 5 ~m 2 1!!, P~m, b! 5 1 2 exp~2b! (l50

m21 bl

l!

(41)

nd @see Eq. ~38!#

w~n, b! 5 bn exp~2b!G~m 1 3 2 n!

G~m 1 3! (l50

m122n bl

l!. (42)

Thus for the function ^g~s!& within the frameworkof the approximation being considered one can obtain

^g~s!& 5 1 1 (j50

3

(l50

m22j11

sjb2j1l11

3 exp~2b!G~m 2 2j 1 2!

G~m 1 3!G~l 1 1!. (43)

Note that optical thickness ^t& @Eq. ~32!# is relatedto the volumetric concentration of particles c by asimple formula3:

^t& 51.5cL

aeff, (44)

here

aeff 5

*0

`

a3q0~a!da

*0

`

a2q0~a!da

(45)

is the effective ~or Sauter! radius and c is the fractionof a unit volume, occupied by particles. It followsfrom Eq. ~45! for the gamma PSD that

aeff 5 a0 S1 13mD . (46)

Equation ~44! can be used for estimation of the con-entration of particles c from measured values of L, t,ef.Thus it follows from Eqs. ~27! and ~44! @see Eq. ~43!


as well# for the intensity of the diffused transmittedlight in the small-angle range that

I9~q! 5I0

2pexpS2

3cL2aeff

D *0

` HexpF3cL^g~s!&

4 G 2 1J3 J0~sq!sds. (47)

Results of calculations of normalized intensities~q! 5 @I~q, t!#y@I~0, t!# with Eq. ~47! for polydisper-

sions with the PSD q0~a! 5 Aa6 exp~21.5a! are pre-sented in Fig. 5 for different volumetricconcentrations of scatterers c and a wavelength of.628 mm. The geometric thickness of a layer andhe effective radius of particles were equal to 4 cmnd 6 mm @see Eq. ~46!#, respectively.It follows from Fig. 5 that the increase in concen-

ration of particles causes a broadening of the angu-ar distribution. This is the same effect as for

onodisperse media. We also present data for theormalized intensity calculated with the Mie theoryor the same PSD, l 5 0.628 mm and n 5 1.33 ~waterrops! in Fig. 5. The small difference between theie curve and our curves at c 5 1026, 1025 ~t 5 0.01,

0.1! is related to the error of approximations in Eqs.~1! and ~43!.

5. Conclusion

The widely accepted approach to optical particle siz-ing is based on the single-scattering approximation.However, many natural media ~e.g., atmosphericerosols and clouds! are optically thick, and onehould account for the multiple light scatteringvents in the retrieval schemes. This is importantor laboratory measurements as well, because someisperse media ~e.g., in the solid state! cannot beiluted.We have found that different optical particle-sizing

echniques7–11 have the same ground, namely, the

Fig. 5. Dependence of the normalized intensity on the observa-tion angle obtained with the Mie theory ~Mie! and Eqs. ~47! and~43! for the gamma PSD f ~a! 5 Aa6 exp~21.5a! at wavelength l 5.628 mm, the geometric thickness of a scattering layer L 5 4 cm,nd different volumetric concentrations of particles c 5 0.000001,5 0.00001, c 5 0.0001, c 5 0.0002, c 5 0.0005, c 5 0.0007.

Fraunhofer diffraction particle size analysis,” Part. Part. Syst.
small-angle approximate solution @Eq. ~17!# of theradiative-transfer Eq. ~7!. Equation ~17! can be ap-plied for the solution of both direct and inverse prob-lems of disperse media optics in multiple-scatteringconditions. This formula was transformed to Eq.~47! for the simplification of calculations of the smallangle transmitted intensity in the case of light-scattering media with polydisperse large sphericalparticles.
This study was supported in part by the Alexandervon Humboldt Foundation. A. A. Kokhanovskythanks E. P. Zege for initiating this work and for hercomments. He is also grateful to V. V. Barun, A. G.Borovoi, O. Glatter, E. D. Hirleman, and V. V. Ve-retennikov for the helpful discussions.

References1. W. Witt and S. Rothele, “Laser diffraction—unlimited,” Part.

Part. Syst. Charact. 13, 280–286 ~1996!.2. K. S. Shifrin and G. Tonna, “Inverse problems related to light

scattering in the atmosphere and ocean,” Adv. Geophys. 34,175–252 ~1993!.

3. E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image Transferthrough a Scattering Medium ~Springer-Verlag, Berlin, 1991!.

4. J. A. Lock and C. L. Chin, “Correlated light scattering by adense distribution of condensation droplets on a window pane,”Appl. Opt. 33, 4663–4671 ~1994!.

5. A. A. Kokhanovsky, “On light scattering in random media withlarge densely packed particles,” J. Geophys. Res. D 103, 6089–6096 ~1998!.

6. P. G. Felton, A. A. Hamidi, and A. K. Aigal, “Multiple scatter-ing effects on particle sizing by laser diffraction,” Report N431HIC ~Department of Chemical Engineering, University ofSheffield, England, 1984!.

7. E. D. Hirleman, “Modeling of multiple scattering effects in

Charact. 5, 57–65 ~1988!.8. E. D. Hirleman, “General solution to the inverse near-forward-

scattering particle-sizing problem in multiple-scattering envi-ronments: theory,” Appl. Opt. 30, 4832–4838 ~1991!.

9. V. F. Belov, A. G. Borovoi, N. I. Vagin, and S. N. Volkov, “Onsmall-angle method under single and multiple light scatter-ing,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 20, 323–327~1984!.

10. N. I. Vagin and V. V. Veretennikov, “Optical diagnostics ofdisperse media under multiple scattering in small angle ap-proximation,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 25,723–731 ~1989!.

11. H. Schnablegger and O. Glatter, “Sizing of colloidal particleswith light scattering: corrections to beginning multiple scat-tering,” Appl. Opt. 34, 3489–3501 ~1995!.

12. W. Hartel, “Zur Theorie der Lichtstreuung durch trube Schich-ten, besonders Trubglaser,” Licht 10, 141–143, 165, 190–191,214–215, 232–234 ~1940!.

13. L. S. Dolin, “About scattering of light beam in a layer of aturbid medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 380–382 ~1964!.

14. A. Borovoi, P. Bruscaglioni, A. Ismaeli, N. Vagin, and V. Ve-retennikov, “Multiple Fraunhofer diffraction in optical particlesizing,” in Proceedings of the Fourth International Congress onOptical Particle Sizing, F. Durst and J. Domnick eds. ~Nurn-berg Messe GmbH, 1995!, pp. 69–78.

15. 2600 Particle Sizer User Manual ~Malvern Instuments,Malvern, England, 1985!.

16. L. P. Bayvel, J. Knight, and G. Robertson, “Alternative model-independent inversion program for Malvern particle sizer,”Part. Part. Syst. Charact. 4, 49–53 ~1987!.

17. L. P. Bayvel, J. Knight, and G. Robertson, “Application of theShifrin inversion to the Malvern particle sizer,” in OpticalParticle Sizing, G. Gouesbet and G. Grehan, eds. ~Plenum,New York, 1988!, pp. 311–319.


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Multiple Light Scattering in Laser Particle Sizing