ATTENUATION OF LIGHT IN CLOUDY MEDIA

DENSELY PACKED PARTICLES

W I T H

A . P . I v a n o v a n d V. G. D a n i l y u k UDC 535.36

Many l ight-ref lect ing objects (soil, snow, pigments and dyes, photographic mate r ia l s , luminescent powder layers , e tc . ) consist of par t ic les lying close to one another . Because of the dense packing of the par t i c les , d i rect light disappears af ter passing through a few of the single layers of such a system, since the mater ia l contains no "po re s . " Therefore , Bouguer ' s law, which requi res an exponential decrease in the br ightness of the direct (unscattered) beams with an increase in the thickness, is not obeyed in this case .* Since the whole theory of multiple scat ter ing depends on Bouguer ' s law in its differential form (as in [1]), the equation for the t ransmiss ion of radiation becomes unacceptable in its normal fo rm. The fol- lowing questions then a r i se : What is the spacing between the scat ter and absorption centers at which Bouguer ' s law no longer holds? What is the nature of the attenuation of the direct light? How can we de- scr ibe in general the t ransmiss ion of radiation in a strongly scat ter ing condensed sys tem? The present ar t ic le deals with some of these p rob lems .

The subject of the study was powdered glass grade SS-4 with an average part icle size of d =14 # . A specified quantity of the powder was mixed with water in a vesse l . A suspension of uniform height was obtained after severa l minutes . By changing the height of the column of water it was possible to create a sys tem consist ing of par t ic les spaced at various distances from one another with a given quantity of powder.

The exper iments were carr ied out in the SFD-2 spec t rometer with a slightly modified at tachment of the type descr ibed in [2] for measur ing the t ransmiss ion coefficients of the horizontal ly a r ranged l i g h t - s c a t - te r ing spec imens . An FMSh-56 photometer was also used. In the SFD-2, the angular divergence of the

impinging radist ton was defined as the angle 27sc =15' , while the t ransmit ted light was recorded at angles of 25t r =20' , 2 ~ and 6%

r ~ : _ .. . . . . . . . . . . . . . . . . . . . . . . . . . . . i Using the FMSh-56 we were able to measure in directional i l lumina- ! tion the t ransmiss ion coefficient of the diffuse light, i . e . , the ra t io

# " 7 1 ! ~ - - - - - - - - - - - 2 ~ ~ of the total light f rom the specimen to the illumination intensity. a, g p . _ . j ~ _ _ _ _ _ _ _ _ - - = The wavelength of the light was ~ =540 nm in every case . With this

1o | / / - - - - II

l/// ........... / i 6 / /

5

/ f f i . . . . . . . . . . . . . . . . . . . . . =8 //"

Z q h, cm

Fig. 1. The dependence of T on h; r /=0 .126 (1); 0.378 (2); 0.63 (3); 0.756 (4); 1.26 (5); 3.78 (6); 2Ytr =40' (i); 2 ~ (II); 6 ~ and

(w).

wavelength the mater ia l of the SS-4 glass has an absorption index of 1.528 mm -t, while the absorption and scat ter of the distilled water was negligible. We therefore assumed that the attenuation of the radiat ion occurs only as a resul t of the interaction between the radiation and the par t ic les of the g lass . The probabi l i ty of su r - vival of a photon A=cr/e =0.7 (or and e are the indices of scat ter and attenuation of the part icles) was determined by measur ing ~ and e in the at tachments on the SFD-2 descr ibed previously in [2]. The diffraction parameter was 0 =Trd/X =81.

Let us now consider the experimental r e su l t s . Figure 1 shows the dependence of the measured values of the ' t r ansmiss ion coefficient

* This is because Po i s son ' s well-known statist ical distr ibution on which Bouguer ' s law has been based is not re levant .

Transla ted f rom Zhurnat Prikladnoi Spektroskopii, Vo. 22, No. 2, pp. 302-306, February , 1975. Original ar t ic le submitted December 28, 1973.

01976 Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. No part of this publication may be reproduced~ stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $15.00.

229

d

\ N\'.."~, r \ \\..K~ -.t~..

1 z y

F i g . 2. The dependence of T on77: 1) ca lcula t ion f rom Eq. (2); 6) ca lcula t ion f rom Eq. (3); 2Ytr =40 (2, 7, 9, 13); 2 ~ (3, 8, 10, 14); 6~ 11); and 7r(5, 12); h = 0 (2-5); 1 (7, 8); 5 (9-12); and 8 cm (13, 14) .

T on the th ickness of the l i g h t - s c a t t e r i n g layer h (by this we mean a highly cloudy column of a suspension) with va r ious angles of the r e c e i v e r 2Yt r ,

For a p a r t i c u l a r spec imen we a l so tes ted both e x t r e m e l y dense or mu l t i l aye red fo rmat ions deposi ted in water and a l so suspens ions occupying a volume of height h. The f i r s t case i s r e p r e s e n t e d in F ig . 1 by h of the o r d e r of a f rac t ion of a m i l l i m e t e r ; the second, b y h o f up to 8 cm. Each curve in F ig . 1 is the r e s u l t of ave rag ing the data obtained f rom 4-6 s pe c i m e ns . Each dose of powder has i ts own over lap fac tor 77 by which we mean the r a t i o of the a r ea of the c r o s s sect ion of al l the p a r t i c l e s to the a rea of the base of the ve s se l in which they l i e . With a complete p rec ip i t a t ion of the p a r t i c l e s at the bottom of the ve s se l , mono laye r s c r e e n s with di f ferent packing of the s ca t t e r cen te r s a re formed when ~7 < 1. When ~7> 1 the p a r - t i c l e s a re c lo se ly p laced with r e s p e c t to one another in s e ve r a l r o w s .

In o rde r to see c l e a r l y the changes in the l a rge and smal l va lue s of the t r a n s m i s s i o n coeff icient , the va lues of T f rom I to 0.5 were plot ted on a l inea r sca le , while those l e s s than 0.5 were plotted o n a l o g a r i t h - mic s c a l e .

I t is c l ea r f rom F i g . 1 that as the geome t r i ca l th ickness of the spec imen for any constant amount (constant number of s ca t t e r i ng pa r t i c l e s ) is i nc reased , there ts an [nc rease in the t r a n s m i s s i o n coeff ic ient . P a r t i c u l a r l y l a rge r e l a t i v e changes in T could be observed with l a rge va lues of ~7 in the region of smal l va lues of h. In th is case the t r ans i t i on f rom a monolayer sc reen with a th ickness of the o rde r of 20/x to a l a y e r of I cm leads to a 10-fold i n c r e a s e in the t r a n s m i s s i o n coeffic[ent with smal l r e c e p t o r ang le s . When h is fu r ther i nc r ea sed T app roaches i ts a sympto t i c value which appea r s al l the sooner whenil is s m a l l e r

and 2~tr is l a r g e r .

However . this phenomenon is not apparen t in eve ry ca se . If the angle of the r e c e p t o r is l a rge the effect of an i n c r e a s e in the t r a n s m i s s i v i t y i s difficult to obse rve . When 2Ytr =180 ~ (Fig. 1), T is constant with d i f ferent h, independently of the p a r a m e t e r 17.

These r e s u l t s can be explained as fo l lows. When 17 was la rge and the sca t t e r cen t e r s laid ve ry c lose to one another there was no d i r ec t b e a m . All the rad ia t ion had in te rac ted with the p a r t i c l e s and had been s c a t t e r e d to the s i d e s . With an i nc rea se in the height of the cloudy volume, on a s t a t i s t i c a l b a s i s r eg ions of space where the l ight p a s s e s through the medium without any sca t t e r a p p e a r . At smal l angles 2Ytr the l ight r e c o r d e d by the r e c e p t o r is main ly the d i r e c t l ight . When this is absent the signal on the r e c o r d i n g in s t rumen t was weak. With an i n c r e a s e in h, there is an angular r ed i s t r i bu t i on of the rad ia t ion , an a c u t e - angled component of the or ig ina l r ad ia t ion a p p e a r s , and more l ight r e a c h e s the r e c e p t o r . It is na tura l that such a r e d i s t r i b u t i o n should be l im i t ed . When the concentra t ion of p a r t i c l e s is sma l l , the phenomena a re de t e rmined by the va r i ab l e opt ical th ickness T =ceph ( c is the concentra t ion of p a r t i c l e s with an a t t e n u a t i o n index ep) a n d b y t h e r e l a t i onsh ip between the s ca t t e r ed and d i r ec t components; consequently, the me te r r e a d - tngs become s teady . If the number of p a r t i c l e s is l a rge , then the height of the cloudy volume must be in- c r e a s e d so that the p a r a m e t e r T can become s tabi l ized with a low r e c e p t o r angle , tn this case the reg ion where the laws of s t a t i s t i c a l opt ics and, the re fo re , the equations for r ad ia t ion t r a n s m i s s i o n begin to be opera t ive co r responded to pa r t i c l e concent ra t ions c of the o rder of 7 �9 1 0 4 c m - 3 . In this case the average d is tance between the s ca t t e r cen te r s is 150 Ix, which is 10 t imes g r e a t e r than the pa r t i c l e s i ze .

230

I, t e l . unit

\ N.

q ' 0 : y, deg

F i g . 3. The a n g u l a r d i s t r i b u t i o n of the t r a n s m i t t e d l igh t : h = 0 (1 ' -3 ' ) and 5 c m (1-3); r7 = 0 . 2 5 (1); 0 .63 (2); and 3 .78 (3).

Wi th a r e c e i v i n g a p e r t u r e ang le of 180 ~ both the d i r e c t and s c a t t e r e d l igh t e n t e r s the r e c o r d i n g i n s t r u m e n t . When h i s d e - c r e a s e d , the c o n v e r s i o n of the d i r e c t l igh t into s c a t t e r e d l igh t o c - c u r s a l though the to ta l r a d i a t i o n c u r r e n t r e m a i n s c o n s t a n t . As a r e s u l t , the t r a n s m i s s i o n c o e f f i c i e n t r e m a i n s c o n s t a n t . F o r th i s r e a s o n we can e x p e c t tha t in the r e f l e c t e d f lux t h e r e would be no d e p e n d e n c e of the r e f l e c t i n g c a p a c i t y of the m e d i u m on the d e g r e e of p a c k ing of the p a r t i c l e s .

None of t h e s e phenomena a r e in any way the r e s u l t of i n t e r - f e r e n c e e f f e c t s , s i nce the l a t t e r , a s was shown in [3], o c c u r a t v e r y s m a l l a n g l e s of s c a t t e r .

L e t us now c o n s i d e r the p r i n c i p l e s involved in the t r a n s m i s - t i o n o f l ight w h e n ~ i s i n c r e a s e d . F i g u r e 2 shows the g r a p h s of the func t ion T =f@) wi th v a r i o u s v a l u e s of h and 2Ytr . A s e m i l o g - a r i t h m i c s c a l e was u s e d in F i g . 2a . To g ive a m o r e d e t a i l e d e l u - c i d a t i o n of the f e a t u r e s of a t t enua t i on of the i l l u m i n a t i o n for s m a l l v a l u e s o f T , a l i n e a r s c a l e was used a long the o r d i n a t e (F ig . 2b) . When the p a r t i c l e s f o r m a m o n o l a y e r s c r e e n and the d i s t a n c e b e - tween t h e m i s s i g n i f i c a n t l y g r e a t e r than t h e i r c r o s s s e c t i o n , the t r a n s m i s s i o n coe f f i c i en t of the d i r e c t b e a m i s

NepP T = 1 (1)

! S

w h e r e N is the to ta l n u m b e r of p a r t i c l e s in the l igh t b e a m ; S is the a r e a of the c r o s s s e c t i on of the l ight b e a m ; ego is the a t t e n u a t i o n c r o s s s e c t i o n (index) of the l ight for a s ing le p a r t i c l e depend ing in s c a t t e r t h e o r y 6n the d i f f r a c t i o n p a r a m e t e r P and the a b s o r p t i o n and r e f r a c t i o n c o e f f i c i e n t s . The second t e r m in E q . (1) c h a r a c t e r i z e s the l o s s in l igh t f r o m s c a t t e r and a b s o r p t i o n . S ince in t h i s p a r t i c u l a r e x p e r i m e n t the p a r t i c l e s a r e m u c h l a r g e r than the wave l eng th , then a c c o r d i n g to s c a t t e r t h e o r y , ep =7rd2/2, and t h u s *

Nnd 2 T - 1 - - 2S = 1 - - 2TI. (2)

C u r v e 1 in F i g . 2b c o r r e s p o n d s to E q . (2).

When the p a r t i c l e s l ie a t much g r e a t e r d i s t a n c e s than t h e i r s i z e the w e a k e n i n g of the d i r e c t b e a m s o b e y s B o u g u e r ' s law, and the m a t h e m a t i c a l d e s c r i p t i o n of th i s wi l l be

r t

T = e 2 h=e_2,a. (3)

T h i s p r i n c i p l e is shown by c u r v e 6 (F ig . 2) . Thus , depend ing on the d i s t a n c e be tw e e n the p a r t i c l e s , the v a l u e s of the d i r e c t - l i g h t t r a n s m i s s i o n c o e f f i c i e n t s l ie be tw e e n c u r v e s 1 and 6 in F i g . 2 .

When not only d i r e c t , but a l s o the s c a t t e r e d , b e a m i m p i n g e s o n t h e r e c e p t o r , then with d i f f e r e n t p a c k - i ng the c u r v e of T = f @ ) wi l l a l s o be d i f f e r e n t . I t is c l e a r f r o m F i g . 2 tha t with any 2Tt r , the c u r v e s wi l l b e c o m e h i g h e r and h ighe r in p r o p o r t i o n to the i n c r e a s e in h. H o w e v e r , t h e i r r i s e i s g r a d u a l l y s lowed down and , s t a r t i n g f r o m a s p e c i f i c t h i c k n e s s of the s c a t t e r i n g v o l u m e , the de pe nde nc e of the t r a n s m i s s i o n c o e f - f i c i e n t onr/ r e m a i n s c o n s t a n t . I t should be noted tha t the r e g i o n of t h i c k n e s s e s w h e r e T d e p e n d s on h g r a d - u a l l y d i m i n i s h e s wi th an i n c r e a s e in 2Ytr, and at the l i m i t w h e r e the whole of ~he l igh t is p e r c e i v e d a t an a n g l e 27r, i t b e c o m e s z e r o . The c u r v e s for h equal to 0 or 4 em a r e v i r t u a l l y the s a m e .

I t is c h a r a c t e r i s t i c tha t wi th low h and low r e c e p t o r a n g l e s (when V a l s o i s s m a l l ) the to ta l i n t e n s i t y of the d i r e c t and s c a t t e r e d l igh t I i s g r e a t e r than I B, the i n t e n s i t y a c c o r d i n g to B o u g u e r ' s l a w . In the r e - g ion of a v e r a g e v a l u e s for the o v e r l a p c o e f f i c i e n t , I< I B . Th i s can be exp l a ined by the f ac t tha t the d i r e c t l igh t has a l r e a d y d i s a p p e a r e d with t h e s e v a l u e s fo r r~, and the v a l u e s of the s c a t t e r e d l ight a r e s t i l l s m a l l . F i n a l l y , if r/ i s l a r g e , the s c a t t e r e d r a d i a t i o n i s s i g n i f i c a n t l y g r e a t e r than the d i r e c t r a d i a t i o n for l a r g e h, and t h e r e f o r e aga in I> I B .

* F o r s m a l l d i s t a n c e s be tween p a r t i c l e s when the d i f f r a c t e d f i e l d s f r o m ind iv idua l c e n t e r s p a r t i a l l y o v e r - l ap , T wi l l be s l i g h t l y h i g h e r .

231

With large values of ~ the rate of decrease of the intensity with an increase in the number of s ca t t e r - ing centers is different for different 2~,tr and h. This is due to the different size of the range of values of

for the formation of a broad light r eg ime . At the limit where ~ - - ~ we would expect that all the regions of the curves corresponding to very large 77 would become straight l ines. The slope of the lines for var ious 2Ytr with a constant h will become the same. The question of whether the slope changes with different values of h requi res special examination.

Since the charac ter of the decrease in light with various receptor angles depends on h, it is clear that the angular distribution of the t ransmit ted light will be different with a change in the size of the cloudy vol- ume . Figure 3 shows the indices of the light intensity for extremely dense and dilute sys tems with various values of 7/. The data obtained with the angle y =0 are taken together . When 7? =0.25 then the distance be- tween the par t ic les in the monolayer screen is la rge . Therefore , there is no overlapping of the diffracted radiat ion fields. The par t ic les scat ter the light independently of one another . Here also there is no mult i - ple ref lect ion. As a resul t , the observed angular redistr ibut ions of the light match one another to an ac - curacy of the measurement e r r o r and charac ter ize the indicatrix averaged over the ensemble of scat ter by an individual par t ic le . As the amounts are increased (the number of par t ic les increased) attenuation and multiple l ight -sca t te r effects begin to appear, and therefore a general tendency to "flattening" of the [ndi- catr ix also appears . The distance between the values of Iff) for dense and ext remely dilute sys tems is increased . It is charac te r i s t i c that everywhere in the ~? region in the study, the sys tem cons is t ingofdense- ly a r ranged par t ic les has a more diffuse indicatrix than in the case of grea ter distances between the scat ter ing centers . It is now difficult to say whether there will be a difference between the angular d i s t r i - butions when ~ is very la rge .

The question a r i ses now from this analysis of whether it is possible to use the t ransmiss ion equation when the scat ter ing par t ic les are close to one another. In the general case the answer to this question is "no." The failure to obey the stat ist ical principles and the rapid disappearance of the direct light lie be- yond the conditions of the formulation of the normal t ransmiss ion equation.* Nonetheless, if one or another photometr ic pa ramete r (in the case of this study the light intensity of the t ransmit ted radiation) is not changed with different packing of the par t ic les , then the use of the theoret ical resul ts is possible . Indeed, with large distances between the scat ter centers the t ransmiss ion theory establishes the interconnection between the radiat ion field and the optical charac ter i s t ics of an individual par t ic le : the attenuation index; the survival probabili ty for a photon; and the scat ter tndicatrix. These optical charac te r i s t i cs can be measured experimental ly, and from them we can determine the intensity of the light f rom the t ransmiss ion equation. With an increase in the packing,, the proper t ies of the medium change, and the original sense of the optical pa ramete r s of the e lementary volume is lost, but once again the field remains constant; the data f rom scat ter theory can then be t rans fe r red to the ease of small distances betwet~n par t ic les .

L I T E R A T U R E C I T E D

1. V . V . Sobolev, Transfer of Beam Energy in the Atmospheres of Stars and Planets [in Russian], GITTL, Moscow (1956).

2. M . P . Znachenok and K. G. Predko, Equipment for Determining Scatter and Absorption Charac te r i s - t ics fin Russian], P repr in t Inst . F tz . Akad. Nauk BelSSR, Minsk (1972); Equipment for Measuring Scatter Indicat r ices of Dispersed Materials [in Russian], P repr in t Inst . F iz . Akad. Nauk BelSSR, Minsk (1972).

3. A . P . Ivanov, A. Ya. Khairullina, and T. N. Khartkova, Opt. i Spektrosk. , 2__88, No. 2, 38i (1970).

* Another point of view is possible, based on the fact that the t ransmiss ion equation is always fulfilled but the concept of an e lementary volume and the sense of direct (unscattered) light changes radica l ly .

232