Graphite polarization data for semi-quantitative estimation of interstellar radiation wave extinction by small graphite flake

Greenberg, J. M.

English

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Wave Length Dependence of Polarization 
.L ,I* 
by Small  Graphite Flakes 
J. Mayo Greenberg + 
The Institute for Advanced Study, 
Princeton, New Je r sey  
Ab s t r a c t 
/2*6 
Y) 
Lu 
0 - a 
n 
n 0 
(3 
The extinc ion by oblate spheroidal par t ic les  whose complex 
dielectr ic  constants a r e  anisotropic in the sense of graphite is calculated 
in Rayleigh approximation. 
flakes (maximum radius N . 0 5  micron) as represented by the oblate 
spheroid model, and not too thick dielectric mantles, the wave length depen- 
dence of polarization may be expected to follow the curve fo r  the wave 
length dependence for  extinction. 
tendency for  the polarization to remain fair ly  constant through the visual 
It i s  shown that for sufficiently smal l  graphite 
This i s  in contradiction to the observed 
par t  of the spectrum. 
.b -1- 
Work supported in p a r t  by the National Science Foundation and the National 
Aeronautics and Space Administration. 
+ On leave of absence f rom Rensselaer Polytechnic Institute, Troy, New York. 
https://ntrs.nasa.gov/search.jsp?R=19660003707 2020-03-16T22:11:06+00:00Z
. 2. . ' b  
lntr oduction 
In a s e r i e s  of papers  start ing with that of Hoyle and 
Wickramasinghe (1962) there  has  been a resurgence  of interest  in the 
possibility that graphite may be of significant and perhaps p r imary  
importance in explaining the inters te l lar  extinction and polarization. 
With the exception of the work of Wickramasinghe (1962) the main 
attention has  been directed to  the problems of extinction par t icular ly  
in the infrared and the ultraviolet ( see  for example Wichramasinghe 
and Guillaume 1965, Greenberg 1965). This i s  a l i t t le surpr is ing 
when one considers that the original motivation for invpking graphite 
(Cayrel and Schatzman 1954) was the deep concern that dielectric 
needles "apparently" were  incapable of producing the maximum 
observed rat io  of polarization to extinction. 
demonstrated experimentally that even a smal l  contribution by 
graphite flakes would be sufficient to account for the polarization 
(a figure of 5 percent  contribution to the total extinction was given 
by Greenberg 1962). The more  theoretical and quantitative t rea t -  
ment of Wickramasinghe (1962) based on the Rayleigh approximation 
to absorption by oblate spheroids with anisotropic conductivity 
der ives  a figure for  the required degree of orientation on the assump-  
tion that all of the extinction is produced by graphite. 
neither in this la t ter  paper nor in any of the others has  the problem 
of the wave length dependence of polarization by graphite been 
Cayrel  and Schatzman 
However, 
ser iously considered. 
The astronomically observed character  of the wave length 
dependence of polarization is  on a t  l eas t  a s  good and dependable a 
foundation as is the extinction curve. 
I The purpose of this paper is to apply the most  recent  optical data on graphite to a calculation which gives a t  l eas t  a semi-  
quantitative est imate  of the wave length dependence of polarization to 
be expected f rom small graphite flakes.  
3. 
It should be noted that the original qualms about the 
polarizing abi l i t ies  of dielectr ic  par t ic les  have already been dis-  
cussed and resolved semi-quantitatively on the bas i s  of incomplete 
data on prolate  spheroids  (Greenberg e t  a1 1963). Recent exact  
calculations (Greenberg and Shah 1965) on polarization by cyl inders  
with a r b i t r a r y  degrees  of orientation conf i rms  this work. 
4. 
2 .  Extinction Cross-sect ions by Oblate  Spheroids 
with Anisotropic Refractive Indices 
Following Wickramasinghe (1962) we consider graphite 
to be represented by an  oblate spheroid whose thickness along the 
symmetry  axis  is 2c and whose width is 2a .  
We shall  be concerned with extinction, absorption, and 
scat ter ing cross-sections in which the e lec t r ic  vector of the incident 
radiation i s  e i ther  along c or  a. 
Rayleigh approximation c ross -sec t ions  for  a r b i t r a r y  orientation a r e  
easily obtained by appropriate  l inear combinations of these basic  
c r o s s -  sections.  
- 
Within the framework of the 
It is assumed that the tensor represent ing the complex 
index of refract ion has  the same  principal axes  a s  the spheroid (a 
very reasonable a s  sumption). 
Clark  Jones (1945) of the electrostatic polarization of an  ellipsoid of 
anisotropic mater ia l  we may describe the polarizabili ty along the a 
and c axes  in the notation of van de Hulst (1957) by 
Following the generalizations by 
2 
m - 1  a 
2 
L ( m  - 1 ) t l  
2 
a c  
a 3 
(y = -  
a a  
2 2 
where  m and m a r e  the complex dielectr ic  constants along the a 
a C 
and c directions respectively,  and the L and L a r e  numerical  
f a c t o r s  depending on the rat io  a/c 
a C 
It is  now easily shown that the var ious absorption and 
sca t te r ing  c r o s s -  sections a r e  given by 
C"' = 4a~ k Re {i a )  
ab  s 
8 4  2 
C(') = --srk I a c J  
s c  3 
where k = 2a/A, and the superscr ipts  a and c correspond to the 
e lec t r ic  vector, of the incident radiation along the a and c direct ions,  
respectively.  
The total  o r  extinction c ros s - sec t ions  a r e  given by 
c = c  t C  . 
ext abs  s c  
. 
. 9  
1.242. 8 
6 .  
4-1Oi 4-1Opi 
4- 7 i  4- 7 p i  
3 .  Polarization by "Model" Graphite Pa r t i c l e s  
It can be shown (Davis  and Greenstein 1.951) for  par t ic les  
whose scat ter ing i s  given adequately by the Rayleigh approximation 
that the rat io  of polarization to extinction i s  
A m  - 1  
= constant 
calc 
kG- 
( i  
'ext/ Lext 
2 c  
ex 
where the subscr ipt  calc is  used to denote the theoretically calculated 
value and where the "constant" is dependent on, among other fac tors ,  
the degree  and kind of orientation of the par t ic les .  
We do not use the quantity ( A m  1 d i re  c tly to e s t imate 
P calc  
the wave length dependence of polarization. 
es t imate  of this quantity i s  obtained by multiplying Eq. ( 3 )  by the 
observed wave length dependence of extinction to give 
A somewhat bet ter  
A m  
'A m'ob s A m  ( X )  = +-- P IP A m  calc 
We p resen t  the following table a s  a reasonable representat ion (See 
Wickramasinghe and Guillaume 1965) 
constants of graphite 
of the complex dielectr ic  
. 
Table 1 
Cross- sections f o r  Oblate Spheroids 
a/c = 5, L = + 7 5 ,  L = .125 
C a 
x - l \P  10 10-1 1 -2 
r 7 
0 . 9  305 32 19 
1.2-2.8 396 41 14 
2 .8  2 .8  4.5 
c 1 
7. 
Table  2 
Cross-sec t ions  for  Flat Disks 
a / c - m ,  L = 1 . 0 ,  L = O  
C a 
, 
x-’\p 10-1 1 
I 0 . 9  1600 170 116 
1.2-2.8 1600 165 65 
l 0 l 2  10-1 1 
-1 x \p 
I’ 
0. 9 
2 . 8  
1. 25 a/c 12. 5 a/c 17.2 a /c  
. 03a /c  32a /c  8 a/c 
8.  
We see  f r o m  the tables that in Eq. ( 3 )  we may effectively 
(a)  “’ . Fur the rmore ,  i t  can be seen la) , c ( C )  
by ‘abs/‘abs replace C ext ext 
that the value of 
A m  
A m  calc 
I- p ,  
i s  a constant within 5 percent over the ent i re  visible spectrum. 
Consequently, to this approximation, the wave length dependence of 
the polarization is just  that  of the extinction; i. e.  it r i s e s  mono- 
tonically f rom the f a r  infrared to the ultraviolet. This dependence of 
polarization in wave length has also been predicted (Greenberg  1962) 
for  the Plat t  (1956) type par t ic les .  However it is generally observed 
that the polarization is ra ther  flat a t  l eas t  through the range 1. 3 
a 3 with, perhaps on the average a maximum in the green. 
X - l  
In figure 1 a r e  shown the observations of Behr (1963) a 
solid theoretical  curve (Greenberg 1962, and Greenberg e t  a1 1963) 
which fits the observation of Gehrels (1961) and a dotted theoretical  
curve based on the r e su l t  that the polarization by small graphite 
f lakes  follows the extinction curve. The solid theoretical  curve i s  
normalized to unity a t  A 
= 1.5 which we have for convenience chosen a s  the point of normaliza-  
tion for the dotted curve.  It appears that  even where  the Rayleigh 
approximation should be valid, the graphite model does not give a 
-1 -1 
= 2, and is exceedingly close to 1 a t  A 
-1 
good representation of the polarization. 
and 2.8 a r e  ka = 28 and ka2. = 88 and one should expect the 
approximation to be valid up to ka N 0. 5. 
The values of k a  a t  A = 0. 9 
. 9  
All of the above calculations have assumed uncoated graphite 
f lakes .  If graphite flakes do indeed exist  in inters te l lar  space,  it  is 
to be  expected that they would accrete  dielectr ic  (dirty ice)  mantles.  
If the  mantles a r e  not too thick (relative to the core  thickness) the 
gra in  would probably s t i l l  preserve a reasonable degree of oblateness. 
Since the rat io  / C ( = )  would, a t  l ea s t  fo r  the longer wave lengths, ext ext 
s t i l l  be determined pr imar i ly  by  co re  absorption (a Rayle igh  approni-. 
. 
- 4  
mation resu l t )  we should find that the wave length dependence of 
polarization would still have the tendency to increase,  although 
perhaps not s o  steeply. 
statement,  but when taken with the quantitative resu l t s  for pure 
graphite flakes,  i t  leads one to conclude that, at the very leas t ,  the 
graphite flakes do not readily lend themselves to  a theoretical  
explanation of the wave length dependence of polarization. 
This latter is admittedly a ra ther  qualitative 
The author would like to acknowledge his indebtedness 
to Dr .  J. Robert  Oppenheimer for making possible his s t a y  at The 
Institute for Advanced Study. 
appreciation to Dr.  Bengt Stromgren for valuable discussions.  
He would a l so  like to express  his 
1 1  
9.  
.I 
10. 
References 
1. Behr,  A. 1963 Pr iva te  Communication. 
2. Cayrel ,  R. and Schatzman, E. 1954 Ann. d'Astrophys. 17, 555. 
3. Gehrels,  T. 1960 Astron. J. 65, 69. 
4. Greenberg,  J. M. 1962 Draft copy (unpublished) Chapter 6, Vol. 
-
I 
7 of S ta r s  and Stellar Systems.  
5. , 1965 Proceedings of I.A. U. Symposium No. 24. P a p e r  
6. 
7. 
8. 
9. 
10. 
11. 
12. 
13. 
delivered in Saltzjobaden, Sweden, Aug. 1964. 
Greenberg,  J. M. and Shah, G. 1965 Pape r  given at the I.A. U. 
Colloquium on Inters te l lar  Grains ,  Troy, New Y ork.  
Greenberg,  J. M., Lind, A.C. ,  Wang, R. T. and Libelo, L. F. 
1963 Electromagnetic Scattering, Proceedings of ICES I, 
Pe rgamon  Press, New York, 123. 
Hoyle, F. and Wickramasinge, N.C. 1962 Mon. Nat. Roy. 
Astro.  SOC. 124, 417. -
van de  Hulst, H. C. 1957 Light Scattering by Small  Particles 
(Wiley and Sons, Jnc., New York),  Chapter 6. 
Jones,  R. C. 1945 Phys.  Rev. 68, 93 and 213. 
Platt, J. R. 1956 Astrophys. J. 123, 486. 
Wickramasinghe, N. C. 1962 Mon. Nat. Roy Ast ro .  SOC. 125, 
-
-
-
87. 
Wickramasinghe, N. C. and Guillaume, C. 1965 Nature,  207, -
366. 
1.4 
1.2 
1.0 
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d I I I 
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'/A- 
F i g u r e  1. Wave length dependence of polarization. Experimental  
points by Behr (1964). Solid curve is  best  theoretical  f i t  (Greenberg 
e t  al. 1963) to observations of Gehrels (1960). 
based  on theoretical  resul ts  of this paper  and is  normalized to unity 
at A-1 = 1.5 . 
Dashed curve  is  


Wave length dependence of polarization by small graphite flakes

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