Light Scattering and Absorption in Pigmented Coatings: theory and Experiments

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/290997480

Light scattering and absorption in pigmented coatings: theory and

experiments

Thesis · April 1997

CITATIONS

6

READS

106

1 author:

Some of the authors of this publication are also working on these related projects: Hydrogen metal thin filmsView project

Structural color of Chrysina scarabs in Costa RicaView project William E. Vargas

University of Costa Rica

75PUBLICATIONS   980CITATIONS   

SEE PROFILE

Absorption

in

Pigmented

_Coatings

c¡npd.Eive

smnris

ot

upF¡la

Disefrññ

aoñú.F¡dltyorscjeoeáddT¿chnology26t

Absorption

in

Pigmented

Coatings

Theory

ai.l Expdiúeñts

WI¡,IIAM

F,

VARGAS

Comprehensive

Sumn

ies

ol

Uppsala

Dissefations

fron the

Faculty

of

Science and

Technolosy

261

Light

Scattering

and

D ssedarion lof tho Deqree or Docror or ph osophy n

so

d sraie physcs pfesenled

ar L,ppsara un vefs ly in 1997

valgas,

w.

E,1997

LshlScare¡nsandabsoDron

tn P gmenred coar ngs: rheory

rps

conpztt

"t

s

núrn6 af

upp:tt¡ tDs:lnd¿iü'

lian

tn¿

fad¡,

ofsúm

dúfünm6¿1261 ro4pp

_uppúta.

_sBNsr

_{554 3s26,3}

d ro describe

oprca

propedies or poymofc

waveensrh

ol rhe

nc denl rad ar

ón

From Lorenz Mie lhoory varous parameierc

d absofpl on by sinote

pad.

es áre cacutared,

ie.

rhe

s.allerng

and absorprioñ cross secrions,

ihe lorwad sóarerns raro

under

lry raclor ard the sns e

pance

phase _rrncrion

Mur pré _{scarériñs efiecls} are iaken into account by ñáans or a nove approach

r oquatoñ fof

semiinlnre

med¡a pofpend cu

a

y lúmináred wirh unporarrzéd fad a

scatofiiq

ór

rhe d,rúse

fad arion

is

descfbed

n

téfts

or

_{senefauzed phasé}

funcrons

and rhe

opii.ár

depth dap6nd6ñcá

É

specifed

by

weighling racro¡s

corespond¡ng

lo

lhé succossive s

approacrr rhe exp cir eva

uaion

or averaqe path 6ngrh pafamerec atrd

loeard

s.allefnq

€1os chafacráriziñg an s

oprcalpfopen es ol poryme¡ c thick rrms.onrañ nq

sio2

r02

(fuiie). or

rio'

Gnr6,

cod6d

sio2

pismanrs are

componenh ol ret

e.tlnce

and úansñillarcá rÍom rhese coal ns§. Good as.semenr belwe6ñ msasremenrs and comp

whenpiomenlpÓfÓs¡lv¡.lakennloaccounl

W liam E

vaeás,

Dápanñenr

ot

MarerásSóénce Uppsara

Unversly

Box 53-4

LIST

OF

PAPEFS

l.

Coñdeñáalion or wardr by Badarivs Coolnq

r.

M.

J

N§gM,

w

E.

Va'!d,

G

A NHson.

and C. G.

Gaqvi.¡.

Br4.¡l€

E¡ersy

5

310-317(19§4).

¡1.

CompalÉon Barween Oplhizal on

Üi.ñarorLishr

Scatednslroñ N0ñ Ab6orbiñq Sphercár Pañicl€s.

W. E. Vál!ár ánd G. A.

Nikhs*.

J. coloid lnlBñace Sol, 161, 497-499 11995)

lll.

opricál Prcp€ni6 o, a P¿n or

sprÉrs:

C.moaison oI

Oifie6l

lh.oies

G. B. Snirh, w. E. vafqÉs, G.

a

Nikrñsn,

J.

a

sd.b,

a.

v.

Páléy. and

op' cs

comñun.ll5

3r,

(1ees)

rv.

uqhr scaGang coalings:

IÍsy

añd 6.1.r

appli*lloB.

W

E.Vae!6,E

M.Lush¡ku,G.A.

Nikl$lon,

dT,M

J.Nihsoñ.

lnr Symp OpticalMálárids Techn. rorEnsrcy EficlBncy and Soar

Emrq{

Csv.Bim xv,

Fr.ibuE,

Gémrry,

17-19 S.pl€mber 1996.

V

P _{smenr Mase} Densiry añd BeIáclivB lndox

oel.m

nal on rrcm Opl

.á

w. E.

várga

.id

G. a.

Niklás*.

J. Phys.: Condens.

[ia(€¡9,166r

1670(1ss7).

vr

Fo

.fd

av€rá96 Parhl.ñqih Pafán6r.f in Radiariv6 Tfans,ef Mod.ls.

w. E

vaoú

úd

G. a.

N¡{áer.

T.

b.

wblilhed in

am!

oDl

vll

FoMald soanefins Rario. and avefag. P.rhlensrh P.fámeref in F.diarive

W. E. Va,gas añd

G

A

Niklassm.

subñfiod roJ. Phys. coñdens Mats¡

Vlll,

ApptEbriry Condirions o¡

,'6

Kub€lkaMlnk theo,y.

w

E. várga3

úd

G. a.

Nfdá.*n.

lX.

rñrensryor Difiuse Fadiar on iñ padcu are Meda.

W. E. Várgas áñd

c

a. Nikrsson

súbmred roJ.

opr soc

ar.

a

tuatn9 Scardrtns parameréE usod nFadiarve

w

E.

vafs4

and G A. Nikásson

Subm ned

bJ

opi. Soc. Am.A

xr.

opricaLPmpedes or

s

icon Psmenred arumina Fitñs

T

TelamchaerW

E.Vargas,E wácké gárd, and G

A

Nikrasson.

Commém on rhe aurhors conubúr

I

Compulalions co¡cem nq oplcárp.oparás or composre S

02ñ

o,

rv

specroscopc measuémenrs, anaryss or rcíedance and tÉnsñillance ffom

poryñ€rc rirñs coñraiñ rq sio2 an

x.

Iheorcr¡ca tomutalion,ándrhew¡ilng.

xr

Fomuarionandappticaton orlháf finsp¡ocádurc, andpadorrh€wdrins.

othef papéfs which havá nor bee¡ inctudBd in

lhs

rhesis afs:

1

.

Chafacláizat on ot _Lighr scaredng ,of composirá

sjo,

T 02 spher ca

Páñcres: N umé rcár Catcu latioñs

w.E

va&as

dG.a

Nktasson

chahefs

un vefsity oJ _{Te.hnotogy Depr or} phys _{cs, T6chn cat Répon}

G pR

e _{Cootins and Cóndensatron n¿sat on}

f

M.J

N s§on,

W

E.Vaeas,

dG.A.Nikasson.

Pfoc SP E _2255, 1e3.204 (1ss4).

3.

opricalPropsdies ot Anodized Alum inu m Siricon Auoys.

SCATTEFING BY PAFTCLES

3.r

Hoñoqañéóus sphercal Padces

OFfICAL PROFEBIESOF PABICULAIE MEDIA

4.2

rL¡any.Fux Radar¡vo Tfansfer Modés

4

5

Boundary Felrecllon Coellicenls

5,2ExlendedHafuTh6ofy

5

3

GenéB zed Mulrip e _Scáhe,ns Approach

NTED POLYMEF COATNGS

61

Oprcal Measuremenb

612

Brlk Felacrive

lnd.esorS

caandTlana

6

ri s É

áñd

rlana (nd

I

Psmanrad samples

614

Samples Contan ns composte Psmenrs

62

Comparsons ol rheóry and Expeímeni

Fifsr ol arr r thank God for His prcvtsion to

ñy

fe.

I am g¡alérut 1o my wte and oúf

ch ld,

shi

6y and José Dav

_d

for p¡ovidins a wondodu

rasl

yeaB

Ths acad6mc

wo*

woúld not be possib e wlhout th6 pefsonalsuppon

slven bymany orher pe6ons. Our grattudero

rheñ

We ar€ speciary ¡dábréd ro our padór áñd his wi,e, En[qu6 Fódfiguez and Manuá rá acuña

venruE vargas ánd o/sa castro

fhis

doóloral rhesis has báén sú pery sed by D r. Gu nnar

A

Nka6son.

_So nany

lhanks ro h m lor lhá áppfopriare qudancé and _{rhe siven} áncouaseñent spéc a y

6bred ro prct. ctá6s

c

_{Gfanqvstorhissuppod}

ns re§oarch _{wo¡k in his group So many rhanks}

andberregádsforweryoneinrh.Marefa opti6cfoup.

ra§ó

rhank Prcr. Jan

E

oreBredl

and pélef Greetrwood ar rhe Depadmenr or

Ensrneéins

_{ch6histfy tn chamáre unversry or}

_fechnotogy,

_rof

_varuabre

discussioñs, and

ior

propaññg

ihe

samptos whose oprica _{propedies hav6 been} am _{s€térúrto rhe univereity} ór cosra Rica, and rh€ costa F can sc enli, c and

Ihe

opricá _Propsdies ol

ereñqrh áñd poañzaro¡ ol ihe

incdeir

fadiaron

ñaúx.

By ássuñ ñq w6 r sápafal

ion, rhe rrndamenrat mode ro

descfbe

rhese

ñcrusions can b6 consd6red by ñeáns ot ñ016 qáñ€ra

boundary

condilion

merlrod

and

rhe

discrele d pore

appfoximarion.

Th6

opl.al pfope

as

ol

ñon hoñogendous

rlms

can

aso

b3

fad¡aron pausm iñp trq n9 on a spécil.

pán.!é

.oud

beñodred

rñcoñparsoñ ro

Ihese

efecrs catr be raken

equarons

for

many

pa lcle

sysrems:

lhe

exlended T

of

mány

pánic.E

ard eveñ

¡

afg6 coñpurors

áre

úsed nuñerica

necassa.y ro coñsidBr ñuliPr6 sca

ln rhis rlress,

wrh¡ lré

indopendeni scallern9 approxmalotr noverlheofelcar

ler

mode.

They are rhe loMard and ba.kward

nlo accounr rhé án9úlár

I

dlluse radialión pare,ns

fom rhe fadiár v6 tañsfor équaloñ úñdef a

s6ñ.

scalré¡ins ¡ar

os

app icáb

t

y

condilions or vafóu§ ñode

s

app

isd n

modetins

oprica

prcpoñas

_of

padicualo

_ñ

famework provded by rhis _seneralzed

murpe

scanerñ§ approach

Tho visib 6 and near infaréd oprcatprcpeñi6s or inrins ca y non homoqeneous

Th6sá ritms conssl or non masneric

deeclrc

or

pherca

_{pan¡cbs hosred iñ}

_a

_weaky

_ábsoúns

poymef

marx. fhá

coarinqs charácredzed by dií616

compafisons

or

rheo16rca

Speóra or coalngs conraining S o¿ (s tica)

Tto,

(¡rañ a), or

ro2

coarad

sio'

parl

model s usad ro calcrlalá co ¡ñared and dtruse componoñh ot retecrance and

rransmirtance.

unrk6 pfevous

apprcárons

igrh paEñeref and

fofrard

scaref ns raró have

y ,rcñ

th6 above noñr oned mul pre

scalerñg

2,

GENEFAL BACKGÁOI]ND

cksrclnd of this

rheÉ

has been provded by

wave

iñpnsins

on á homose¡eous and spherca padicte tiosred

nan

nrnre

non

absorbinq

marx,

rhe exptcir sotu

coofdnar€s, werc oblained by

Mé

n rhe

bágnnns

of

rhs

cenrufy

(Me

lsos,

L iárrerd 1991). The comp át6 rorma i§m is usua y _{rcrered lo as Lorenz Mie rheory}

due ro rh6 prevouE work

cared

our by lhe Dañish sctenrisr Ludvq Lorenz (Krash

1Ss1).Thoiñcdenr,scatafsdand _{nt6ma ete.lícándmá9n6tc,ierdsaÉexpanded}

Dué 1o rhe cont ñuiry ot th6 ranqen

rh€ Irerds ar rhe spherca

unknowi coerrcients invotved

nthserpansioñso,rheinrBmatandscareredáecrc

(slraron

_1e41)

_a

s _s vén by Debyé (1eo9

Many €xrensions of rhe Lorenz-Me rheo¡y have béen esrabtished in

odef

ro

con§idor exr n.uon by: coátod

sphefcá

_{pancres (aden and Kerker}

_lssr,

_Gú

_6r

1e52

Ftrrtr

1ee3), homosaneoG sphsrtcarpádicles _{áñbedded n an absofbinq}

Lakhlaka

ree5,

ouinten and

Rñ'akk tss6) sphe¡.a

padces

srppo

ng

onsirudna poáfizarion waves (Fuppin 1e75

Md

rssr)

_{charsed padictes tBohren}

and Hunr 1s77), hoñoqo16ous mannetc

pañdes

(Kork6r

rs33),

and

ayefed

sphércarpadc6s (Bhandaf rs35, sinzq and ouinr6ñ 1s04, Kaiand Másso ise4).

KhmssKrcnis

anaysis (kú atrd Feiske 1ea6 xine a¡d Gfeentrers 1se2), sphefca

(Maheu

d a. rm3)

proxim ry or a prane inrerlce (v do

and

vdeen

1se7)

notr

spherca

pafl.6s

(¡\¡adin re§3),

sprrs6 pañrcñ,rh

emedricaryspher¿ar (FúIer

lse4aand leesl

or fesúarincusonstvdeen

eia

1ee5 Hir

€1a

lees)

m

tus

(La

and

s*ss

ree5)

and

noirnoar

6ri6ds

(Dewirz er

a

1ee6)have al.o been srud:ed.

Hoñog€neous and coabd sphsrcdd pances have been ¿óns defed by sorvinq

n

spheróda

.oórdinare. _lasano

a.áno

1979, Coofáy áñd

C[ó

1994.

The

rended boundary condion

üf

malrx úsed as an

alemarve

approách

absomron by spherc dár pañi.re§

(wábrñán rs63 r07r

and 1e7s)

rhermdrx

s wirh shape§

fore

comptcated

thar

sphefo da

oñrcrced resurls n rhese cases (wscombe and

Muqnail930and 1933, Muqnaiand Wiscoñbe 1936and 1939)

appfóriñalón is

a

b61rár ápproach

b

characrdrze

adicres _{lPufcerrand Fennypa¿ker} ie73, Lumñe

U 1994,1á.zkandBooker1996 Laczik1996),

Dópend6nr sca ering efiects on exlinclion proped 6s IÍom rwo.panc e syrems

Rayro gh.D6bye.Gans

_{approxnalon loaole}

19701

fay

oprics and LorenzMe

afared padctes (wans 1ea0, !./anq

erar

1e31).

só útoñs 1o Laplácé equalloñ by assuñ

i9 ñur

poláf expansions ol rhe porenr a s

tcrrdy

and

aÁroos

!s30, Ruppin

rs624.

and

rinh

ereménlñerhod§ rcrud¡is

(Hodeáu an,r Ma'1n

1ssE.

arb

eiabrsrrins

f

spharicar wave funcrons (srein

rs61

ctuzan 1s62),

wh.h

are o'lhoqona sourions or rhe uedor

Heohorz

equarron lBayve

1e3r)

rE

comdsdoB

or depa

ñelanc

sphof.d

pances (Brunins and Lo

'e7r

a db)

fhe

errecr oi dependeni

aisons

o, rhé e

e.ri.

and masnel c I e ds, has

besn considersd by Fu 6r (1ee1)

d pances (Pererson and

_Sr.n

te73 and je74

slroñ

1s74). Due ro rhe

very

arse coñpuráronal

requreñénrs

mosl spocirc

appicarons or rhis éxrended ap

padcre sysráñs

(cdz

sr

ar

et

at

res3)

n rhe

cas6 of

scaterns

ún(s compos6

Lorenz Mierhéory is a moEveEarire approach (BoGh6se er al

ausoos 1932, Mackowski 1991 and 1994 Fúnef 1Ss4b, Xu 1995) and r has beer

app ed lo

ñádiuñ

széd c usrefs o

(ou

enandKrebs1se3

soieo and Niktássón

lser,

Kfeibs and votm€f res5)

w

hout con§idárrng d6p3nd3nl sc

d opucardáplh dependences ot ro&árd mún

pe

ñ

n padcútare media coñrainins monoszed and

Endomly distibuted sphefca

padc*

(Har16r1s40).

a

s milaf approach

was

arer

sven

byfhe$

ns

_lsso)

ñ6

Hañé rofmalismhasbo6náppuedfordescfbnsthe

ansurar dgpend€nce or th¿

difúse

nlensly dué ro scarerns lrom ñonodispefs ons

(wóodwed 1e64). as w6

rhem (smañ 6r

ár re65)

o

rer applcatons ot

Hane

rheory are rctared

ro

iqhr

sca ériño by cers (Larmer 1s32 and 1e3E.

Nsson

mélhods tor sizins

É

oda

pádcres (scrr n abres

qd

áñd Gknef

rse5

schnabésséf

erat

1ee6), ard

§drca

melhodslordacualinssparia

momenEorñu¡plescareredphorons{Lúrom6k¡st

Anorhe. §ourc6 or basc concepts whch ar6 impticir jn lhe conrdnl or ihis thes

s

s rho rad ar v€ ransrer theo'y (chan

BeÉ31933), wrh emphasis on phoron _trañsood

rh,oush

¡ghr scaredns and

ahofbng

mareras. For ptane paGr6t _o€omeúies

e _{translef équarion hav6 b€en obrained whén}

charácle.z¡ns oprical pfopeñ e§ or páncu are med a (Hore

and

ri6ñ

re3r,

F6nker

resr).

rh

schEbr

(1s0s)lákins

nro ao.ount two dtfiuse compoñsnrs or lhe

hdaron

tietd,

soing iñ opposire

di@l

ons. From rhe €dialvé ránsfer equauoñ rhe lwo I ux ñóde

of Kube ka ánd Munk (Kubetka _{and Múrk 19s1, ]<ubá ká 1943t has been obra ned}

(shiñau

1s73). andrhré+ (Ryde 1e31. Fochman re73).

rouf

(Beasreyerar 1s67 Maheú er á1.

ls34),

and

sx,tux

mod6rs (chu and chu

a

1sel) háre been rofmual6d

mpiemetrrad (Richads 1s70. Múd 1s71 and 197,' Wh

hey

1s71)

rn tho n6xr secrions wo winrocus on thosé aspecrs reraréd ro

c app carion coñs¡d6r6d n

ihs

rhesis

.e

vsbe

ó

o

operó

orpqn

q.eo

polr

d

od¡

a.

3, SCATTEFING

ferracrive ndex wtrich iñ

pfncps

can be

ermaled

iiom rhe corcsponding

buk

ágñéric

teds

iheir uous ar rhe pan

_ce

súfraca (Lora n er

!r

1e33).

an be áxparded ln an apprcpriare basis of vecrof

waveluncloñs

lnlh6faf

feldzone

.e

Iarawayfomlhépance sphercaroe.rc

úsed ro spécry rhé

spansons ofihe

e¿dfc

and magnarc

felds

nthscase,rh6

báss ser o,

vedof

wave

lúictiois

s given by soúrons ot iho v6óror Hemho[z

equaron (Fuppn 1s32b)

Fbñrhebomdarycondlons,thácoeffcenb

nvotvedtn

nrema and scalldréd fields can b

sphorrcál

_Páricles

The simp esr case or a iinire

pance

geomorry cofesponds ro rhe sphortcátone. smbedded n anon absoúns med

a.e waverdñgrh ot rtre impinsns Édiar

on

Fóñ

masner.

feds

ánd app calion

ofrhe

boundafy (Eohren atrd Hurman 1e331

ai

mvi(nx)v;(i)

r,,rx\,n(,nx)

m\u,(mx)q,rr)

(,(x)v;(m,

h, =

y!@!y¡ql¡Yl!Yí]]]])

_.

₍₃₁₎

'rJDx)(.(x)

-

ñ(.(x)v;(mxr

wrh

n=]23...

rh6

an

(b¡lscardrriq

coefrcenr spe.res lhe ferat v6 .oñribur on of er¿ (masneric _{of rañsveEe}

_ee*.1mode

ro

$e

fesp*r

ro rh6 ársumenr

ii

pa _{Énrhésis vn} x ) añd q¡( x ) 3fé rh€ so cal

e¡

B'ccári

Bésss runcrons

inoss

1972)whi

Hank€ ,unorons respeólivály .0.

,ui(rl=x

r(r). and

(i(r)=xhfr(r).rho

.labe

numef¿a ávauarion o, rh€ soate

seriunc¡¡oi§

(aden 195r, Dave 1969, L6ñ¡z

reo.

Bohf€n r9s5l. ouf cñpurarions havo

b*n

cafíed out by applyins also,ithms

erab

Eh6d by

wscoñb¿

(1930)

l¡6

szo párameler fangá

corespondn9lo

our

app icál

ón s

appfoximál6y b6rw

asofrhñ§§uiráblorof compulatións

nvovñsv6ry aQes?épafamerefs(w.ngáñd

van d6 Hulsr

r0§1)

Ffom rh.

sne,ins

c@ficienrs

rony

olh6f pafarerofs

€n

be

q,

¿6úfpri{

d enincriói

cos sd'ú§:

oscÁ.

o.e,

and

o6n=o$a+oae

(Kérké¡ 1969. Evans and Fournlú 1990),Ihá asymmelry

fadol

q lDsbyo 1eoe GouBsberel

a

1e33)

th6loMad

scauorins fario,

r.

_lchyek

1s73

WsooñbeandChyoklsTT),rhescárr€.n9añprrudás sr(F)ands21[),

and

rhe sinsré-p.n¡cE

ph*6 r!rcli.n:

p(u) _{wherc u} is rhe cosin€ ol 'he

scatedno ¿nsre

Thá 6 n9 e páriclo pháso rundon

scaref€d rad

aron

nrensly 116

nuñárca

6válualon s rarhd lmo consúming when

m6d¡um or lárce size páránelere are .onsidor€d. Apprcrmato exp¡ess on§ or rhe

af6

áváfábr6

(H.nyoy

.nd

Gf.eñsr€'n r94r,

Rerdds

áñd

rrccomÉk

19ü0. coméhe and

sh

ks 1992) wirhin rhe LoGnz.Me theofy th6 rrgorous sinqr..pad¡cre phas6 luncrotr can b6

wri

en in

remg ól

rhe

scate,ns

amplrudes

óf.xpañded

n

Lásándro

poynoñiás

Pn(u) ov6n

a um

_nal.d

wirh unpol.rzod radiari

niE

₌

_^--l

s

ip)

_'

+

stt¡)

_':)=

:

o¡P,,(¡1)

ic

pt.n.

wáve

is ¡ñpi.9i.9

¡+o.e,

s i3 ¡h6 polár.ngl€ m@

and the 5cálld.iñq ampl

_lud$

are qiven by

*=3*

lÉ

,""=,/:

*J

§,rp+

irq+

)fwNrm

rN,+

\,,v,,"rB"lJ

f

co+rr.rio+

r

r+q¡

ll

'

where §pq s lhe (rcnecker delra

a",,

=

(2^:)

t

(2n+r)

Re[¿"

+

b"]

,

w"-=

n"[",";,+r,ril

_,

v,.=

R" [""bi, + b,,;,1

fh6 inleg€s

lnñi

andliñihave

been eva úared by

rnnr:¡1*rFm,m+

I(+r.+

Q..

=

r:/{¿)¿ 12Éll

L

!rd,l[u!rq][ore,

t*r"i

_]/

l'

)'

I

]I

I

j

s

an odd

numbs

añd zéro ¡r

j

É

an ávgn

numbar.

Furhamora,

_¡Dn,=l

ifo<n+m

i<2m

ánd

vnnl=l

if

1<n+m

i<¡¡+l

and

rhe

phase runclion, as

qven

n

Eq

13.2), s noma ized ro

1 ro _r=1.

oo=OfJO

bá ng rha 3¡ng o'páÁicre

(3.5)

ámpirudes and rhé éxpansons n

rerñs ol

Lágendre

pótynoñas

(Eq

3.2)

The

ot lho s¡ñqá.padce

phaE.

_rú¡croi

_has

_atso

_been

Anolhá¡

scállédrs

paramerer which can be catcrtared

rom

rhe

s.a

erng

66ffc

enrs is rhe asymmelry rácróf, q, which c _siveñ by (Gráarf er

a

1992)

p=,rotrn '-2rR- q, b:0" , "..k¿i,^

w

h 6=xrosa. ll can b€ sas y proved rhat dr=3goo {Gouesber er a . 1933).

(a)

l1

(b)

a

2

4

6

3

10

s¡;,+iEk,.e"e.-,r-m?0ñgfreéfrcvryds,lh€eüb,!:05{Ém

For ñ0 n- or slis hlry absorbins spherica pádc es, rhs s ze paEmstér depé.dencé

ol

rh6 noma iz6d

6lincr

on

cr63

séclion, oe:r. dtsplays a backsfou.d oscillalon

d upon the main osc¡ndion (chyrék

rgrca,1975

and 1976, Parm6r

d

a.

r$9)

Thé ámptirdee or rhe peaks in rh

¿nd in the

ipp6

srudúfe

ñdex incfeas$. and rhei widrh§ d

den.e or lhs

votrmerc

s.á

eing cross sedotr

c..,/v=3Q{¡/ar

wirh

C.,¡=¡¡1Q-.

osc rar¡on and rh6 rppre

srudure

Ef¡acrive index ncrcases (chyek and zhañ

1s3s)

rhe

narcw peaks iñ rhe fppte slru.lure, _{dúá ro ñ0rpho osy-dependenr resonan.es in hi9rr órdef}

padal

su'1a.e

erñefrary

in

rigtrr

scarerns

by qrowns wáler dropets lDobbns and Ekrund

re77)

and by

6ns

oúÉ

y revrlábd

(ud

drops (ashkin and Dziedzc 1e31)

ne ro charadei ze rhe

fomhoogy

dep6¡d6ñr

(HunEr er aL

1s33

sr 1se3

vdeen

leesr because or rhen d rcrradve index ol m¡crcscopi. panc es (chytek

atrd

s[varava

1e33, chyrek or

ar

is33,Ráyda

]se])

rhepEsrc¿orlherppre

ion

foss

secrion s supprcssed by noñ sphefica

shapes s 26 d¡§rriburons, and ofonratonar avohsins (Brymr and Ld

mtr

1e6s

Ryde and

MdiÉvic

ree4) rhe

fesonancos iñ row6r pañrárwav6s havá wid6 peaks whch afe srpeimposed n rhe

drucrurá has baán inrdlpreled as

bong

pr0duce.r by

iñleleren.e

berween rhe

rr¡aded and ransñ xed isr'i in

'hene

_peakshflsro

ioead d

_áQef

{'bn

_{vahssof thaszepafáñersf wrrenrhefelarverefEdvélndéxór}(Lmk and Yans

lsel)

rhe

pos r on or rh6 ,61 and hiehed

s a so dercted pf _{maq, or láfoe.}

scaed oscl ar on. Trrs oplica ProP

byrhs

nrefércnce sru.rure (crewer 1s41 Dévofe atrd prund 1s17 Fáxvoq re75)

panc e§, have been opr m zed

ú

ordef 1o obla¡n h _sh d rrus6 opri.ar r6r6crances nóm

'hem

(Hadins 6r

a.

rs60

paps

r)

¡

'he.as

of

ro thá

iñaoñary

pa'1 0r rh6

parrr

nd6r

(Kdhwáf

áñd

Pks

r s67)

rhe

absolpron crcss sed on É

prcpoionarb

'he

pañich rc ume

_Gudfe)in

rh6 case ot wéakry (h _shry)absorb trg pad.res (Borrreñ

_aid

_Névfr 1e33l

G ven a spherca padicle

rhe snstá-padice phasé

funcloñ

n terms ot rhé sca

erng

ampirrde§

chyek

" liIIp

n i,.o;.':tq

F

";

Á1é-

_{¿ srará.¿}

_Fq"a

₎

ré"-which is eva uared over evan n16ge6 and

^

i¡i

r,r

cmi r:i+rrm rr

¡,

]mnl'.]ffi

Theorherwaylocarcúlar€rheroMard sca éftns raro

sfomrheexpan§onófthe

s nsle parlicr€ pháse rúndon in L

(cra*

er

a. rs57

orchard 1969)

(3 ₃₎

-l ,:e...

_^,.e

.l

p"..,,

_a1-

-r

_',

l¡

n,l

r'1,

rd

si=ororn

sven (M!ds6t1and F.hards re72)

mpinsins eleclrcmagneric fadiar¡oñ is úavet nq aons

ai

aúrary

tho 6xpans on or the sinst6.pañc e phase

sen6rárized by us n9 rhe addirion rheofem to¡Lásendre potyñomiars

I

p6frecr

y

isotópic

imping¡ns qiven by (orchald 1s6s, Fechman te73)

r,uru

P¡!,!

dB | -

_I

_

=! t.:e.ú

, a.",J.t_o.,,

p,.

lrre

sñal

pañice

mir, ha[ or rhe sc

isphofe Thé slngé.pancle pháse

_I

L

I

L

1o

a

pane

páfpándicuaf

ro

rhe

ncrdence

direcron

(see

Fis2);

hénce borh

o.

dol,

rc

ecur

ro 0so Beyond

ths

srar

pa'1icre

here (r(éfk8r er aL. rs75, Macke

d

a.

ress), and

o" and

ol

cañ ,ak6 vá res

bdw*n

o.5o

d m ly

3.2. Coáréd

SPhéricar

Pánblés.

and

a

compex relfacrvo index No, .oared

wrh

a concenlfc

sphercat

ayer o,

rhckn6ss

.ro

chaa.rerzed

by a

masnero

leds

árá éxpanded n the thrce rcgioñs and bolndary óondirons are

appllod ar rhe rwo sphedca nleflaÉs.

fh6

6care¡ ns coeficiárb

a€

_sñen by (adeñ

and Ke¡*er 1951, Nishtow6fard Richadson 19a3 Katséf and s.hwe¡qer 1993)

.

r;!+

mrtr"

mrr,f,L

)

rf¡

₌

¡,.

_I

^;4')-

A" _I

^;"

+

virrn\)q¡r\)

mq¡(\)q,

(mr)

dv€

ndexórrhg sh€n

r=2nfNrlo

isrhe padi.re

t,

fti)

₌

B^[

=

B^l

mq"r\rx;m.

("1\

x,

rn1i ]

n'u

j(ny)v;(my)

m'r.(ñ

y)v;(ny)

ñxi{ny¡dlmi)

-

m\t¡(mr

(ny)

gy;(!r-l)vrm

vr

r¡

_{vi (n}

y)\1

_,(my)

where

ylx)=

rylx)

spherca

Neuñann runclon

y¡(x)

m=Ne,N,

s'he redñe €lrádñe

ndexorrhe

our numef car evatuaro¡s

y Eqs 1310thavo

b6arcáred

our ac'ofdns ro

dfterenr mareíals (vafsas and N

klason

rss3)

For non.orwBákry absoúng ñarerrars rhe sze parañeror d6p0nd0nc6 or rhe

peaks and va 8YS

are srrified 1o lowér or larger

§z

rerractive ¡d ces a nd pañicle compGi'io n

(vrss

a nd N k

sson

1 se3 )

pad

ces

(P _úchino 1s31)

wásdrcpds

(Bhand{

1e36) iñrdmaréñ6rsy d6ñsr0s o, nor absoún§ sphere§

llock

reeo),

and

dem's(l(aÉsda

1se4). n

.onnécrion ró fadiar ve coo ns appicalons rhéfma om llance and §o ar rclrecran.e

or

powrhyene

pgñenled

lo

s co

rheorerÉary (uójás

d

a . 1s33,

neka

r s03) and exps

mdary

(cEnqv

r

and

Hunds

reTE

verysma

ñela

coaleddeedrcpad.es na

have been slúdied

i

connedonro

arrsofpr 0ñ by surfa.6 pasmon ex.llár¡ons, as n

deedrc

coded mdaric

panc4

(Mcúd

ar

leal)

D116érerled Éorpapscórlañiñsrirariumdoxidocoalodsilcondioxde

he

psme

s

re

and

ib mmpc

rion

(Hs

d

ar

1e$).

I

has

'he

dcaly

(Kdkér

6r

a.

1s75) and expermenrary

lFenne

and

Fsdah

r0e4)be.n

§hown

lhd $a'reins

propen es or padcu are med a conrainins

ve arso b66ñ coñsdered lKallawár and Nood

3.3. sphsroldal

Parliolés

Lisrrl

sca[efng

rfom

a

snse

ho

pa'1.6

hás beei

a.h (Walémai

re63), as

welas

rrom

expl.l

r975)

Nuñeficar

been

appred

in

d scaróred by a sph6rotda pa,1c _{6 tMalsumua} and

_sek

1se1 and 1es6), ard rh6 _oeomerica opr cs rimr has aso been.ón§ dersd

in

rhá case or

very

ar§6

sphercda paricbs

(Lock res6

a

and

bl

Eonsared

paíicrás orroúnd padicles wirri afrne surrace roughñesscoud bá round in ptsncnr dspefsrons. ltdép6ndsonpafaméreBcharacrerzinsrhespectcchemicar pfocess

ú1 iz6d lvisca and Mar¡ev¡c

1s75

KawarrAh and

Mdilevh

leeo

Buininq

d

!.

1ss6).

rhe mor

s¡gnfcánr efied or

crufé (Larmer 1e8o)

o-g

E

o-g

+

áa,

^ffi?

₄

Fis

3 dsprays rhe siz6 paramerer dependeic6 or rrre votumerc sca erns crcss

and spheroida

padces

havins eq

ll

ee3)

have been

app ied. an afrhmeric ódentationa

poafzaron dfectons

hás besn c

lras

recenr

y

been

aveÉses in

ar

anaryrica

,áy

(Khr

ree6) fhe

shapé

of

sphéróida

pances s

chafadefzed

by the so ca 6d

belween

séñaxes

For slishry 6onsded

padics

1as

rwo)

'he

moEhoosydepende

f

iiterl.rcnco

ddcrure É

r

dsplaysd Fo, mofe eonsaléd sph

(Arano 1e7s, asano and

saro

re3o

aMx appfoach is fesr cred by coñpurar onar

coñsdériñg

vefy

áf96 andror

eon

Bymem§orañayricár

rhe

@¡se

ol

apprcabiiry rias b

d oul (Mishchdñko and rravis rss4, Macke er ar

1995, MÉhchenko

d

a'

1ss6)

z Mie rheory rhe

raNiron

ñárix

T

becomes

diagónárand rs s6manls áf6

dredy

prcpodonaho rhe scarefns

coefcenrs

a¡

Loroñz'M 6 rh6ory wri6n n0n sphárica pancés né6d lobe.onsidercd

sorur¡ons ol rhe Maxwel equarons

añd áppy

rq

tanslátioñ oPo¡alore

rúncloN

ceniered ar each one

orl

I

scalr€riñg ánd absorprion propedies

ol

arse páñi.re crusreE (Pele6on and sr,óm rs74

Rme

ly

mo,e

súlabe

exlended

r

ol

40

sph6rcat

pad.es

wefe fepondd (Mackowski 1se4 Mackowsk and Mshchonko lse6)

Sphercár padrce sy.rems can b

Lor6ñz.Mre rheory (cérardy and

a6ro6

re84

Fis 4 dep

ds

nuñúm

EsúG

nz.M e rheory or cérafdy ánd ausoos llhrs rheory

is desc,ibed n papor

I

_).

a

rwo _{pa'1].e sysrsm has beeñ coisidefed, áccofdtns ro} rhe seomerry dspiayed n Fiq.4b. T

pancás

s denored by c. Th6 rc

ar

_{tn Fs 4a}

_cslarv

_s

coñpáfed ro rhé vo

uñoric

scade

(Fs.rb),

_no

_ssnfcanr

_{d,terences af6 disprayed wh6r consderns anorher}

_€lher

sepafarod

pad.

e (.Ao=0.50)

as

_s

in rhe case or quire close paíices.

s aggrdsaled ensembtes or

poyrytsne

are: padicres coñpurálions rrcm the g

oú

and compared w rh ñ6asúremenrs Nesig bte dependoñr

scadeíis

éffsds wefe

obséR8dj

whle lor

coroida

pani.6s

_rh€

inreracr¡on

rfe*

(our

an and sr ef l essr

E"A

lt

L__,

l.

^_L

accounr dep6ñd6nl

s.áfEfñs

srfe

(vaadan

d

a

reTs.Madá

1es.4)

MN'

app

É'ms

or €d dive ransrermodes are based ón the iidepeñdénr scariedng approximaiion

whch smp ties

rrro reqúÍ6d ñarhémarcar

Ioma

sm

(Niklason and Efksson rs33, Nirsún md NkBsn

rev)

G ven a psmenred mareia, rrcm a

when

the

dependBnr scarérlñg

6fl6ds w

I

modlry

ih

oprica

degfeeóIsóaref

n9andabsofpriotrbyaspecicplnicuarañeduñ Thsr6a

paror

árpicrry

decoupred

tfom the

errecr

ot

tre

ñk¡rsc

panice

absoFlon

srard.a

merhods bas

absorbns

sma

pancés rrave beetr dev6ed by rwe6ky

_le75).

acr ve ndex,

ie

by deneasins scatreús lrom the

ú6

pan.ulár6 ñ6diúm w ldscrease (B€vin and

Bfówñ

1962).

rr has

aso

been

erperinenlaly

esrabrGhed

thal .cfeasnq

rhe

medum

_tby

nreasns

the

p

1.e

cmÉ

Eioo,

rhe

efrsdva

absofprron wirrincrcase due ro

the

ncreased aveEse

n emsnins irom mu i pre

$dbf

ns

ere

s(B

s

re62). rhe

affscr or pancre con

ñd 6ffécrv6

scarefnq

prop6di6s o, piqñeñred

ñeda

has b66n

erp¡.ly

consde,ed

(churclri

et

al

1e61b, orchard 1e6e

PhilpsandB

meyor

leTE

a§ thé

panice.óñcéñtálon

nfese

_te

'he

tarem

s

on

dsrese9, bú

I

rrás beeñ erabisried lhal beyond a cenain pa'1c e

.oñconrrarioi vaue these

prcped4

rai

b

d4re#e

(FNs

1s¡

Ftwdsand

res3)

lr

s rcducrion or the pañ.re

or dopéndenl scaref n9 arrecG when rhe pad

_ce

au*

(Drcen

and

ren

rsa7)

By

bking

do

1o an exlended T malrr romalÉm vaGdan et aL.

(1e33) lráve shown

lhal

beyond

rhe

¡dependenr scarérinq

ápproxnaron, rre

eiie.tivé

a

enuarlon o, €lectomagñelc radial otr p,opásarlns rhrctrgh a

pan.u

are

ñed um rs

rronqy

depeide

on

r

Ihe

Mó

paáñáte6

panicbs (Ho(él ér á1. l e70

á.d

1s7!)

3y ompadns erporio€ñra ñ€ásur€ñ.nE or

rhdófálca

predicrion§ obrán6d

lfom

§óviñg

numeri¿.ry rhá fadiarv6

rEnsfé

iñd.pendenr

s.atr€rns

Browsl€r añd Tién _{{1§32) háv6} concLud€d lhal al

v.ru6 o,

crio<0.30

rh.

in,rEnce ol depend6d scatefinq

efÉrs

on lhe oplicar

prcpedi*

ol pa'iÉuláre

mdia

m.y be

iñpoftnt

lhi§

inlru¿nce is áko

ddemin€d

by lhé

speitu

varues oI rh€ parlicle

sizá oa.ámeler. By as.umnq a droñbohed,ar distibur oñ ór

lo

dislnguhh

balw6án lhe

Lndepénd.nr (d1d0.30) áñd dopended (crtd<o 30) scan€f ns rcs¡ñ.s i3 _siv.n by

-rL

or!

fl

l

i!; rhe p.nicb rcrume

lturion. fhis

cdrefiM hás

been áppri.d by vafqás áñd Nikras$ñ (pap€r

v)

ro varidato

üén

d*.iptioñ

or

ñodél ánd aesumnOlhá indápondedscall6ing approximálion,

Frór

Browráránd

suspénsions

áñd

pigmenred

páids

apprcxinár6tchaderi¿.dbyo.5<r<5.oaid lo+<f <10'r,c.nb.ñod.nádby

ápprying

rh.

indepádenl

soated

may b€ dr.prayod ar rai!€. pa.ricle _{vorlm€ l6crioñs}

T.OPTICAL

PfiOPENTIES

OF

PAENCUIATE TTEDIA.

Givoñ.ñ inhomoqeñoous mediúm conr ring ol ándomry d

ríbuléd

inclusoñs n

á

noi.

orw6akly ab§oÉlng

maríx

¡ho opr¡cal prcp6ñi6s dápÉñd ón rhe shap.¡

sze

nctus óñÉ,.§ wel as rho roon¿eñlfations. tn rhe

limir or v.ry

sroll

in.lusions (mdarlc nam.pañicbs in a

c.Emic

ñarü

is

¡

ry¡cál

exmpl. (G.úqvir

ard

suhmn

1976». rh€

_a¡icál

_{prcperios aG siven} in

r.ms

ot

rhe

_{erb.riw dierdd.}

ruñcriú oI

t'e

mare.irr, wh cl' .an bo obrained rrcm sflecriv.

ñedium tháoros (Marw€l.Gamorr

1904

B.uggéman

1935

Wood añd

Ashrofi

r

ai

1§31 A§pnes 1932,

Nk$son

and

c¡ánqvr

r$6,

MaoMillan ér al. 1991,

Shalad

1996).

lh6

basic áppric¿rions or

ottelivé

éñed

rfh

ñoiallt

(Góñ.2

_.t

ar. 1e3s.

M.h.ñ

1s33)

ofióñÉ

(Gmzérand Madin re72

_a.d

1s73) spháicát ircruEions,

md

rhEush

lsm'h

1ee0

Mbisedar ree4)

EÍecrvá ñedum approa.h

,or coared panc Bs (sihvo a

lser),

as wen as asqregated _{sysrsms tBorshesé el} ai

rsaT and leel)

rhe

pañcre slze ranse

i

wrr ch erlecrve mediúñ theor6s are

appicabe

s

festicted

rdqu rémenr or crraraclerzinq rhe

mGi

be sáÉr ed (N k

Asn

and

cfaiqv

d 1es4) orhaMise a

pan _cle

_diete*ic

,únclioñ d6[ved rroñ qúañluñ

m6.hañ6

pr ícip

s

wou d be requ red

_lceDd

d

ar

A rscenradvañcéacóoñprishad

irhsquásÉralic

imilisrheanayicsoLuton

ol

ro accomr rhe inreEcron beiwéen pá s (Fldchik é1ar 1992, Pa6y ér ar. 1993 paper rrland afays oi sphefcarofc y

rdrcárpádces

(Rerben

d

a

rss5

vasov

d

ar

1ss4).

whú

caúd

Arcrher bas¡c achevomenr

s

rcar6d

ro rho

ol

arblrary scrruclure

fom

a

mulrpoar

appróach

ly

proved thar Ihe Causius Mossoti equaron feñáiñs vá d iñ rhe cáses ol §pheflcalry symñerrc pá r disrfbú¡ons luncrors (Fu ár

a

ree3

Fu and Fesca 1ss4 and rees).

as rhe sze of rh6 iñcusions becomes afger scatrering efecF ñusr be rak€n

r

o

(Bohfáñ

1e36)

Coroid!

rrorinq

ard

ábsorbing

fadiar v6 ran§rar ,omur.r ons (cha

6ss 1933) Fof norhom096neous slabs

ofpláñ0'

s po arzarion ot rhé p'opasár ns ¡ád ar ón (Nole

rs33), andrakiñs poárzalion into accounr lHanssn añd

rrays

1s74) as werrss

coisde

derraces (Lam and

hhmaru

1es4). Feledance propenies or rarge powder §yslens háve

aso

beei

Mandeis er

a

1es0l

d

orhefappfoaóhés áre based ón Ih6 coñcepr or

enersy

sxchaige

borween

abeóins

mdsias

(kúberka and Munk 1e31, Reichman re73. Máheu er aL 1e34r.

Drlusion moders

aÉ

appropdara w

(Slei.ke and Sháph6rd rs33

1993). Monre Ca o simúralons afe qene¡átmáthods which have a3o boen úitzed

lor doscrib ñs rhe pmpasarion or Édiar on rhroush pa'licuraré med a _lzaccan|

i99i,

Seruanr 6r

a.

1994.

Eddows

el

ar 1ss5). rhei

main disadvanrase rs that rhey

afe

usuaty rme

descrlbe

lhe

propagalion

h

partcuard

meda

(Boññéf ér

ar

1s37 Gandlbakhche

d

ar.

rse,

d res3).

slárislica

merlrods rof

(Luromiskiá

_a.1se5

_Kornko

_d

a

_ree6).!¡d

lrácra apprcaches

_lschm

and

Kumr

10eE,

h

erecedrybeendevsd

4.1.

Th€

Radiariv¿

fEnsfer

Fomuta

on

nfnt6

medium cons

dnq

ot randomtv ánd

homoseneóu3ry d

*buréd

pan crss, perpendcu

ary

r umirarsd

wrh

unpo arzed ereclroñagnéric f adiarlon

descf be rh e absofption a nd erastic ig

ft

sca éftns ot phorons by thé36 pan c es

we

dent

ol

timá and azmúrha¡

angó

and ñéstecr

polafzarion

_efecrs.Ihé

tfañspo d 6 qú arion becomes _lrshiñafu 1973)

p{,u=

_¡,

_¡1*

r

wh66

r{d+p)z É'he

oe¡cárd6p'h, d md

₀

é

inrrñic

sarsno

and absuoron

ooericienls p6r

unr

ensrh or lhe med¡um

um nalod plané inrefface orrhe ñodtuñ,

Fcoso

where0 srh€

poar añq6 measufed Ifom the z coord¡na16, and

l(f,¡l)

s th6 inrensiry

otlhedtfus€

yntng v6dor afound an iñrniresima so id anse

c wav6 pfopasar _{ns through a mndom}

_ñeda

_win

be padiany po árlzed

Ield wóu d be énrptuarry po adzed

rhe wav6 pora'izarion wrtchanOo randomy (shimaru 1s7B) A tourdlmensionat

inre s rod rlefe nliar 6qu ar on shoud be sorvad when exp ciÍy.onsidednq poarizar on

errecls (Kaflawar

1ee0 Lañ

ánd

shim4u 1ss4)

By assumins rhe independañr

nd

_0.an

b€ evauared

lron

rhe corcspondnq

ros

secrions, csla/v and cabs/v, añd thé pan. e

Nañery

a=fc-"/v

and

_l]=lc

_{É¡v. w}

hn

the cónren or rhe

and Mont6

caflo

rechn ques,

p0,[

_{) is}

a re arive

pfobabi ty or a phórón be ns 3can

by

asstrmins isotropic

sñqe.pad

PlÉ,A)=oo

anayllca

obraiñed

{Ks

r07r.

Efmover

a

rrr5).

Nuñéricar bchn qres ba§ed oñ

G

sian

háve baen used ro so ve

lhe

radá

pe'1ss or panled rayere (strafey ánd Kuñiromó

.

1s331

Ftr

meda conrañiñe

ase

panic*.

anse appoxLmaroñ lKuea er

a. rs

rss3)

and

ow order oxpanEions o

Lógendfe poynomials (Yúrcvch and Konyukh 197s)

1on.

By coñsidefns trish ofder

éipansoñs,

a

se¡erárizatioñ 0r rhis proc6dúfe rsadsro rhe so caned PN appfoxmaróns (Lú er

a.

r§e2, Karanjaiand

ra

ukder 1sez).

4.2.

Many-Frux Radiárive

Transfer

Modets.

Fbñ

thá Édálion inbnsi'y,

(ru)

lorardifuse _{6diárotr rluxes sotrg} iñ

to

afd

ul(z.u)du,

and

Jr(¿)=

lablshéd in phenoñ6noLooca ways, rhrouqh romurar¡ng rwo couprod neardi,raránria áqualioñs d8scrb n

rdándr¿

(sch6€r

1s05, Kubsrkaand Munk re31):

^1

,'

^1,

{;

=

rs*«rr,

*

s:,

!U!

=

(s*K)

¡

sr,

lasa)

(.4 3b)

prane.parand gaoñerfy

R¿1l

_lig!§¡11

I

wh€re

_&

k rhe

dfi66

cfred €

of rhé báckiñe or subsráre

h

s rhe srab rh ckn6ss.

S

and

K

are

e,recrive

scatlorñg

and

absoE|on co6ricenrs per

uñir rensrh

a=(s+K)rs. ánd

b=lir-i

tr añd _{B @rnkwo}

ri

1e72

Mudsef and R¿hads 1e72, Gare

r§74)

rhe

Kubeka"Munk mod6r is very mú.h used

lor

describins dffuse fél ec

absoÉ nq mareras

(rshmaru

_rsTE

_a

¿ompám6 a

Kubeka Munk modet has

beei

_{siven by vafqas} añd N kasson (pap6f

v t

By

po¡enrs _{lh166. and}

_rourÍux

_{modets have aho}

b63n obrained fom thB radiarve lraBrsr 6qualion (Fechñan

rerc,

Ehmaru 1e7s)

r6rá1ed ro opticat propanes or: psmoñtsd ptas|cs (saundeBon _{1s12), tayered}

sysrems _{lKub6ka 1es4) paink}

(Kewinshac 1s6e

phi

ps and Bi meyer 1e76),

mnehs

(Eqan

d ár

1s73), dmosphefc

dur

(Lindbfls

1975, Lindbe¡q

d a. rss4)

boogóalissues

_landefson and

parsh

]ss1, soyrÍied

and Fukshansky 1933

ch6onseia

1990, _{Fukshansky 1991}

_orémán

_{and Erakke}

resl

Yamada and

Flmm

1esl)

fibfes and vroorlrhiébaud añd Kneubuh

reml

n4anic

powdss (E.khofi

_d

_at

_reeo

_{chrryera.1es5),}

_acry

_ics

_n

_iqrids

(zwnkersdar

!eeo), tarse

móon

panics ensenbes

_{(sa$e tss3)}

paper(Heñiet

and

Fiqda[

]es4),

co

ods

_lRyds and Maljovió res4)

coarings _1orc

era

1990,Gundeeral 19961.

absorbinq coarnss

lli6

basic _{mode wás esrabtshdd by Eeasrey er}

_{ar fs67)}

_and

efs

(rs34

añd

re36)

ro obrain

o&tciry

rroñ

¿o imaled añd dlruse componen

romuarion ls based on the sout o

S=

r"-nr,.

S=r"+r,,"

*4=

in',

«,

*t",.*«,

o¡)dr¡+ocúl!+(t.6r)úr!

$=e0,.*fr,

o.ro,,

E(r o¡,)o

r¡

ocar!

(r

o!)(r!

where

i

s

an

erreclve averase palhenslh paramé16( rhé

aw¡as€

pathensih ompaf6d 1o rh8 cor mared ones, andod is rhe

rowatd scalle dns ralio lo¡ d

fuse

ratlialion Moreover, I. án d .rc déñ olé r he rád arion

beams

qoins

n

m

o, rhe

scaréfns

lnd

absoFlon

parañ616re

assúñed

4Inir;

n a prausbe

a

The moderhas been

áppll€d

snsnsivoy mor

y

us

i9

lho approxiñal on od=úc, añd

as

n rha orisina model (Alkin. and E

lheye,

1963) assuminq approx mate va ues lor th6 avorage

parhlenqrh pa,amerB, or usinq ir as a rilr nq paráñersr (Niklason and Edkson rsBs,

N krasson 1991, N rs§on and Nik asson 1991, W rson and Eck 1994. pap6r xr).

nraron d€rerm narion melhod (cuLdl et

at

ieeo)

N k asson (rs37) has

_I

v6r a characrerzarion o, app cabrfy coid riois of rhis

Iou¡

d

6n0

_iand

«1{d)s

Ii

ins

paÉmdss

Mode

(á

_{ro .}

_rrá

ó'

_ó.ó

bóé'

and

usnsq

as

arriñs

páráñerer

l

lse2)

Prioriolliis

pafañérár and rhe

lonrafd

s.ar

parems. a rsw appróxiñárá ñerhods

id

rya uaths

_t

have been púb shed (wans

d

a

1e3e Een

D

drset

4,4.

Ferrecr.nc.

.nd frá¡.ñlfiance

coñponénr

wá @nsilbf a páfriulare cdrins

depcir.d

or a rhick

subshb

P6recrly rlar

tilñ

Édrat¡on

l¡6

3ysrám ol d lloEni áleqúaloñ5 4.5 has be.ñ §o ved by aeasley et a

(1s67) and rvah6u añd cowo*á,s (1s34). s€cond

odéf

dlráronrrar6qua¡ions,of rh6 n in a

cúois€ mbrion

in

rems

ol rh€

lollding

a,=E^/F[¡+r«(É;]

.

A:=dqlo"+§q(

r -odF

(o.j

aj=d1

(

La( r6,,)+F(

I o")l

((1-óJ)

.

,ilh

(=o+$

n

$6

onidioñ

oooíicienr

p.r

unir

lensih.

By consid€

¡s

enérsy

cóñsé-árion a¡

coáriño iñ¡é.lac6s, rhe solurions

or

rh€

3ys¡eñ

ol

dilréfenriar oqualions lead to 6xpricir expr.ls ons ror oollihated and d fuse I uxos (Mudsen ánd

Rióh.'ds reTr), and férrgcranóá.hd

ráñsñ

anc. compóñ.^rs (Mahou ér

ar

1e34).

thé

spdular ánd

d¡liu*

rolhclancé úmpon.nls áre _9¡v6¡ by

inlodaoos atsfilm

andlih

a

r

rcspo.rive

y

ánd

cÉA[(Ar+r:Ar-Rr(Anr.ArL

(4.3a)

c,=lla,

(R¡Aral)l

cdsh(a,h) +

_[AfA5

R,A,)+AlRsarar]

sinh(A

¡rl

_.(4.8b)

Muriple boundary

rcrocroñs

ñ rh6 subsrare have been

iaken

ñro accounr by ñeans or rhe erfed ve rer ection coellicie¡ls tor

co

mared and difiuse Édiar on at the

cosh(A

jh)r{Ala3

Rrar+alR.Ar.Arl

whec:

¡.

(r.

₎

i,

rhe

cl

ú'ion

merricied ror

co

mared radiár on át rhe trrerrace ri

n

brÉb

Gubdrábár),

¡l

(¡l

)

is

'hé

f6' ác'ioi coéncé

rofd

derrce

r

m

sbsrde

_Gubs'Eb{

r),

ad

4

s 'he

rerredion merric

s'

rof

difius

radiárion

al

rhe iniedace subsúale

fiim

By means ol lhe

oplca

rh¡ckness or rho

púar6 rhe

eíéd

or

bd€b

abelpr on, _{E\ beins}

the

avEragé pálhlongih parameler

lor

dirluse

radalon

propagalng _lhrou9h the

specrcva ué ór13 oñ rhé opr calpfopenles ol Ihe coaring The l6n§m lla nce .ómponeirs arc qiren by

,i ,t

e

::,,,

.

_{b,." ,,^}

,¡,.+,^

¡

o ¡

o,.

- .ta,P-r

whsre

c(a'h)=cosh(A,h), s(Arh)=sinh(a

h), and

By app yñ§ rhis rou¡fux fad arve rranst6r mode eva

!aliig

rom rhe Loreiz.Mi6 eñsrh 1d and 0) and the

fo

ald scarerins ralo uñd6rcorimared n. dent rad ation (ú.) rhé rorwa¡d avefase parhrengrh

rc denr

fadia|oi

G atrd odt

iiom

rhe

e.ended

Hade rheory

reradañ.s

and

rammram6

_{(&. !=R\L\!+\}

_¡

and

1

!¡r!;t

_{)!!+T¡j r)} cáñ be

m

tab

absofpr on .óeficien

I

par

úñr 4qrrr

of

'he

b _{ndef, 0o.}

b

rhe

nrnsc

_absoDlton

eñ

_!iB+lli,

(Kmbmo

el

ar

1r3s)

4.5. Boundary

F.fection

coérliciénrs.

rn ofder ro .arry our compurarions fe arons lor

lli6

16l scr

o¡

.oeffc e

berwé6ñ rrm and subsrale

irom rhe

Ffesrc

romu ae

_lsh

maru

]se1) fhe

boundary rel ecl on coerrictenr ,o¡

difiuse radiar ón can be esrimated rrom (Ryde

ts3i,

Judd t e42)

(0)=

I'

l

,J

,l

o"e

,

l

r:j!

!4L

lce

.

"El

r

si,,ro/+

nd _fL are th6 perm

hvry

ánd pefmeab

tyoflre

reaiive

ro thE

permrv(y

añd

pe¡ñéáb

ty

o, rhe

ñodum

from whi.h rho

¡adaron

_{has come, respocrivety as}

compex cháfacrer or

.!ú

añd

plr

vefase efecl or lhé padices on rhe rciracr v€

inder or rhe coarinq, we evlluara

olsmarpadicevorumefactions

r,

thas

b66n srrown rhar rhe efecrive pefñillivily

and

peh6ab

ry cañ be ¡s ared ro rhe roMa'n and ba.kwad scarsfinq ampñudes

,

sl1)and

sl.r)

fáspdclive y (Bohren 1e36) one

¿.r=¿-[r

+

(3ir/2i1[s(l)

+

s(-l)

I :e-

_11+y],

¡1rr= l

+(3f72i1

s(r)'scl)

where x=2¡11,o is rhe lree space

malfx marária. Fof

sma

sphárcal

o ven bv (Mdwel-Gamet 1s04):

size paEmeier, ánd

Éñ

is rhe

pemxvtv

ol

the

pañ¡c á3 lrré 6tred ve pefmin v

_v

or llre medium is

(416b)

l

+

(2/3)

_1l

(4 14) hav6 bá o n genefarlzed to

..*.-l-#

r,.,=ri*l

ln

rhs

way

,rom scr andUd¡,

ileflácds, can be eva!aled

(

r5)

approximaled

by.!!=ۖ(l+,

_rusl

rik. Eq

(414a)

foleoflhepanC€pefmitvlyandpefmeabililyEqs

scalrárns on rhe

opl.a

pfopedes

ol

párlicu _{are ned a}

qven the añquraf

dkrbuton

of rha

pars ons

or

rhe fadarion

been súnmafzed by van de

Húr

(1e30)

rhe

s sillñ fcanr y

_simp

ed when rhe padicre phase

cared Heñyey.cfeen e

i

prrase _{rún.rioñ. whch}

kéd

,o

vard scaréirq

parerns

(Hgnyey and

Greensren 1e.10).

.ófospondns

ro srccessve

s.axefñs

ofderc

háve been devÉed

(GÉbq

añd

cohen

1e75).

we,0.6

on

approach* ba,ed

on

udtr

of Lo,enz'Mie rheory rhe sradns po ¡r is the romatism ésrabtished by

rañer(re4otrór

doscfibns

rhe

anqurar

_{depoñdences ot}

prcpasarnq

rlorhe

roüard hemÉphere

(rhe$iie

1s5o,

woodwrd

les.

_smad

d

a

i965

Hec

r9a3).

Fof

sem

ñ'in ro panicu are ñedia conranng

¡on

absofbns rheory predicrs a saruraron vauo

fadiaron goinq nro the

lotuad

trem

rsárúfe has bé6ñ f€cosn zed as

ar

a¡omarou3 _{pfedicrion o, Hadetlheo¡/ lorchafd}

Fnd, we hav6 éxrended Harletlhe

rheory our extendéd

vecon

do6s nor d spay

ar

anoma ous behav or ror

opica

y

nón

absoóng

padicres e:pans o¡S lhé elie.live arcláge p

undef anrsóropc fomafd difiuse I

rigofousry rakon inlo account

the waveonsrh pañc e

sze

and

rertu've

ind*

dependercs

or

( md

6d when

carrylrs

od

compurálions ol opric

rwo. rhree or fouf flux ¡ád árve ransrérñodets.

phase _{luicrion a¡d d,,use Bdiarion nrensry n Lesendró poynom}

phase runcrions ésrabrshed by

Hare

and rhe oplida deprh dependeñcé É obra ned

lbñ

6rprcr sorurons ro the radiati

G ven rhe sing e-padc 6 phase Iu

p(rr').r

scoNena

tdériñsa

p¡!,', =*

This runclon descrbes lhe scállered rad ation patérn ror a

_sd6n

ncidance fandomly and homoseneoúsly d sríbuied ndáp0ndenl scau6f¡n9 approximauon

rlre

phase

s evanb can be _seneEred rrom pr

(!.lLl

dp

r

_,ip

r¡,Fu

=I>

whe¡á Nk is a

nomá

zál otr conslanl

aid

the evalualotr ol lhe ¡nlesrar3 has been done bv indúdon IÍom _{nlú=p(!)ranoo.}

:,

(5.r)

^/

(,*,,iiÍi!i,u)

5o

.!

as

dspayed

h

F§s,

'vhen rhe sca

efn§

ofder, k. reaches arqe vaues rhe senera zed phasefuncronrNl[) bocomes morc

soropc

.e

fNlr);1/4r

whén

kJ-fhis

expecred behaviof has also

beei

obsótoed _{wtren cá.utarins photoñ momenr}

llom señéraized random sark módes ([otinko er ar. res6)

Therora

dillusoinbnsrysons ntolrréromadhenspherewaswf (€nbyHade

r(z.u)=ta(rr$)

wherc áách

_Q(a

k

a posirvé d6f nire quanry whch spe.res th6 16 arive amounro,

f'=.0.

¡o+Pr

_Ol

l:k

=

oar,

(a+P)

a,

,

*irh

k=,.3

w

h

'he

bomdáry mndi'iore

_Qt¿=0)=0

_Hane

_{heofy has been}

_succe$tu

_wh6ñ

cerr suspensons (Latim6r r932

lid

ts83).

as

ndióar€d by orchard

_1is65)

Hade s

rrreory d sptays an anomalous behavour in rhe case ot opr ca y

rh.k

ritms conraiñ ns

a sarurá on vatue of lhe tola rans

By lakinq inro accounr

ihar

re re auvé nrensiry ol th6 uns.arered ád aion ts given

by

_Q,Fe

(a+P)z

_fie

sorulions ro rh

q*1,1=

@41.

_r"+»

(5.5)

averase páihensrh paámerer ot r

raken 1o

unry

a

pausible

_afd

phenomenooqca way ró remov6 rhs

it

behavor or

Ha

els

lrráory

É

averase

parhens

r

paramerers Ekwrh

kr.,.3.

Gren

rhe

m6ms or qkwe

hke

nto 4count thá parhténsrh or rhe

d

use kdiár on re arive ro

_he

parhengrri or áñ unscatercd

beam

Fo, rhe ro&ard and backwa¡d

dÍus6

fadiaron g ordef rhe averasé párhrengths áre fespecrivety

rr0),lp

By usins Eq.

(s2)

rh6 ñrosrarroñs

,ri,z"*',".

qls"

'

-

-

r Zn+¡

,

t

a'tr¡t

1r*,,

n,,e.O.;t/

I,

t"

I

1,"

,|

I

ihe

asymmelry

lacror

gi

belongs

lo

lhe

sel

ol

coellci€nls

i3' =.o"

can

be

easlv

evaluared by exp

,h.

**a,.

p",h,*s,,,

p"-*e. -" *"-,

ás

ke-

wh¡ch s lhe váúé cha6c

ons¡der

ony

lhe avefage palh0nslh parañEler

rerared ro rhe rorwárd diÍuse rad ar m,

_il'')=ar

rhe

díirre

rar

.qmÍos

_@uprns

rhe súccessivd scát6riñg ordár§ can be wrrlen as

'L

*=E,,"a',

q*

(d+P)a'

wi¡h ¡rE

loudary

@ndirions

_Q(Eo).

Tn€

Íct

ofdef

sdftn

can b6

e.iry

obúnéd.

andilcanbow¡nen

n loms ol rhe oplioa dcprh

a(o

=f

¡

e

C, r).'i

(d+0)li,

lr

ln

ordí

io

solve tuf mé

ñoí

*anefins

ord66

w.

wfi'e

Q(r=p(,

e1a+ptr.

fhé

pfavouB d

lefontia

equarlons 5 ab .an rh€n b€ €xpressed a6

e¡l,=Frl r+'+'41+>a,rl

e,:.

..e,§

.l

11¡6 boúndáry coñdrons

plz=0):0

hávo bá€ñ mpósed

nio

lh.

dirr6renra

equalonSl0

wo

r€lat oñs ,or lho coéfiic .ñ15 Fr and Gj r. Tnáy á,6

*+

(+r){d+E)

p(¿)

= d <r I

Ilr

(z)

wirh

t>2

éd oñ rhé

¡

€sraring

O6za!

1073)one cañ 6oq

th¡i

rh€

o.,,

_=."

_-!-[.,.,.Y

o,'

'

I

q¡

_c,.r

L :i

_l

c,'=*,ft.

c'.-

'

wrh rho iorlowins iniria vá u66

F,

=.o:

.

c,

?=.+

r,

{5.r0)

racbf

mérhod

(Bendé..nd

whch are obra ned by sovnq 5.r0 expriciy ror rhe r rst and s6cond s.arer ns óders

nd* Emh4

arse

vaús

_ir+EL,;2

aú

¡he prdvjoús

anisotop.

§orur¡ons do nor appy du6 ro rrre

divefsns.oefice

§

G

_{r. we}

have

srablry

ol

the ansoropc

soturions

h

arsured

I

íEr j

>lor

Fo,

sñárú

drrer

be disrinsulshed rrom rsoúop c onss. when

_Érrir.r;,

d

prh vaLu6 afe so urions o,

Eq

1s.10)wlh

l+r

I. isotop c dslibulion or hish scafrer

rrio high ordor

scatafns coeil.ienh,

o(z),

are.haEcrefzed

by a véry smoorh

values

once eacrr

_n(r)

has

be

mmpded

rhe

mrespondns

_Q

.)

m6l

be

ápproprar0 wéighr ns radors ln rhe expansion or

rhe loMard dlruse Édial on inrens iy

ory do.s nol dlsp ay

ái

of medla.onlain¡ns non absorb n

erof

in Hanelvpe rh6of6s uhich

is ñprbr

h

Eqs (5

4)

d (5 3)due to lr'e

nesbd

¿

tlescrbe

n lhe

net

The expansioñ ór

lhe

Io

ard dtluse itriensily in lerms

ol

scatlerng ordé' .onribur ons rEq. 15 3ll can be rew

fu.É)

=:

ci(z)

Pi(rr)

, wirh .i(z)

Thd

'n'69rared

rotua,ddlrse

_'¿d¿ ion sqive¡ by

l-:l

rLr!)

Llu=

2tr

¡,(rl+

)

\,¡(rrA.

;

Numa Ér eva udions ol

(r)

ra

deplcrsd n F q 6 wheré r

=

2!r!

_4n

t

orong¡h

Bnge

ánd

1h,..

répr€son¡alÚe marorials

have

b6n

chos.n:

(.)

rutib rirániuñ dbxidé piqmenrs iñ polyerhyrene. (b and c)

xde

mrdx,

d(d)

ron oxide

panoes

nar(B,

E

E

=!

5

!

E

3

{

é

2

t

!

!

g

E

bddb}EBffh6padrcovouñetadoñaidnee§pú.

Th6

béhávior ol rhe basic Nar6 Lheóry

_s.eady

dÉpáyed in Fg 6a.ln cásár ol

msda coñ¡aiñinq wáákry absoÉns panc

*

_lFss.6

b ánd c) borh ñoders prédic'

d&feásiñg varues oI iG)

{há

f ircÉes

béyond

r=¡.

búr

ríe

b¿.ic Hánd

$sry

ovad'imdeslheintoroi'y

lhe d¡lluso radiaron

patem

s peaked roMald and

ls

eíecuvé avefase parh¿nqlh Ffom rhis exrended Nanel theory

súmmafzed in §eccions

43

añd 4..41.e rheloMard averagé parhensrh pa¡ametef

L

and rhe

fofw

d

$ahsins

6'o

6d)

can bá a^ary|caIy compured

fh6

rófwafd aveEse path _{6¡ slh} parañero r is sive n by _ipaper

vil

'.I

i;1i1"

El'r

=

r +

4f!+it

!@h.

wherc rh6 6xpanE¡on 5.14 rias ba6n us6d. Gveñ á

pad.e

irúm nared with a dinuse

s 6rio can be obrarcd rroñ

(pap$vÍ)

l,

l

I-l

evaluallon ol rhe nleqrah readslo

l,'

*

l,

ri

2í

A"

-ri,

_2i

:,

I'**'

¡*ii@".

Erc¡(f)

P¡(u)

Prn(u)dI

equalro zefo ir n

Moréover.

_r""=l

(lñ+

) r

d

rñ+m s ¿n odd numbe,

n=$

il

crl"*

3

[.+.+1

,(j+rt)]

(1tr:1,):(2nr2j)

!@

₉₅

i,@,_

whefe ln/21means rhe nroera pan of n/2. t¡ rhe case of a

cr(r)=o

rorn=]

,.3

.

and c0(r) wi

cmsq@

y

6d=oll'

Frs.7 disp ays rhe oprica déplh depen{renÉ or

i

and

odro,

dfierenr padicre rerarve rcrácrive

n

_{rhe rerÉdve}

_ñdsx

vaues

cofespoñd

to

rhe

md visbe

waveeñorh ranse

cons dérád: (a and b) añofptroús s

(c md

_o

ircn

_eadcsin

ar.

!

f

Beyond a ransenr opridaLdeprh ranqe a

eiuG'ion

q _{vaúe É}

_dÉpkyed

_and

'

soes

rroñ

vau€s c ose ro 2 0 ior §máI

pances

ro va ues ñ6af 1.0

tor

arce on6s.

The Ioruard

s.areing

fatio disp ays a weak opt

_ca

deplh dependence rr h᧠va úes

We w ll cons der pigñeñred coat ns

renlhs (n rhe

ne

nrBred)and á rew hundrad (in rhe vkib€)

ofE

and

dd corespondins ro rhá oplicarrhicknes

o

he

coarns q(a+p)lrl

and

adlG+p)hlwrhhasrher

mrhickness,inth6iou.ruxmode

prevousydescrbádro

ry has been exlended by takiñg iñtó accounr rrr€

th-é d

lereri

scatrdins

ódefs.

Now we rake inio

accouñr rhál a pad 0r th6 rádiar oñ oñéfsinq afief k scaredns evetrh, s gong iñto rh6

From rhe radiative transtd 6quaro

háper

rx) have

ddvéd

rh6

Ey

ám ot

dfiaÉnria

equarions invovins rhe rerárive

amoúnls

or dfruse

rcdarron

qoins

nlo

rhe

rofwed

_Iali¡

and

bekw

d

_loi'\

heñÉph6res, af1ár k sca&,i¡g évánrs:

4ai"

_{= E1')(d+0) 01') + €1.ra}

_o"

dg

=,Ei'l'.B)

al'!á-En.itl

$=

e

r*p,

Ell

lr-ll,l"al,

a

_elll

E1

\ú+B)

01 'j +

(1 r0

o.

_,

af!nÉfllr

_"illl"

afl

+ d,

_¡il,