Computer program for calculating normalized Mie scattering function

Shahrokhi, F.

Wolf, P.

English

NASA Technical Reports Server

N A S A  C O N T R A C T O R  
R E P O R T  
LOAN COPE RETURN TO 
KIRTLAND AFB, N MEX 
AFWL [WLIL-2) 
MIE-SCATTERING FUNCTION 
by F. Sbabrokhi and P. Wov . * I ,  . . ' _  ,; 
- . . . .  -. 
'?-, , 
Prepared by 
THE UNIVERSITY OF TENNESSEE SPACE INSTITUTE 
Tullahoma,  Tenn. 
, 1 ,  ,z  
, \ .~ 
L, P '  ,. ' 
*I. -,.,,I; & 
. 
,;$& .:.id; 
I : .& ; <. ,,v; 7 ., ." *J  .: - : 
fir 
. _: 
. ,  
<. 
. ,.. 
N A T I O N A L   A E R O N A U T I C S   A N D   S P A C E   A D M I N I S T R A T I O N  W A S H I N G T O N ,  D. C.  A P R I L  1968 j! 
i I  
f 
i 
I 
I 
https://ntrs.nasa.gov/search.jsp?R=19680011758 2020-03-12T08:32:57+00:00Z
TECH LIBRARY KAFB, NM 
0060~09 
NASA CR.- 1022 
ME-SCATTERING FUNCTION 
By F. Shahrokhi  and P. Wolf 
Distribution of this  report  is  provided  in  the  interest of 
information exchange. Responsibility for the contents 
resides in the author or organization that prepared it. 
Prepared under Grant No. NGR-43-001-021 by 
THE UNIVERSITY OF TENNESSEE  SPACE  INSTITUTE 
Tullahoma,  Tenn. 
for 
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
For sale by the Clearinghouse for Federal Scientific and Technical Information 
Springfield, Virginia 22151 - CFSTI price $3.00 
~~ " ~ 
MIE-SCATTERING  FUNCTION 
F. Shahrokhi*  and P. Wolf + 
The U n i v e r s i t y  o f  T e n n e s s e e  S p a c e  I n s t i t u t e  
Tullahoma,  Tennessee 
ABSTRACT 
For a compar i son  be tween  expe r imen ta l  r e su l t s  and  Mie- 
s c a t t e r i n g  t h e o r y ,  a series o f  c a l c u l a t i o n s  w a s  made t o  f i n d  
t h e  n o r m a l i z e d  s c a t t e r i n g  f u n c t i o n  f ( Q >  from t h e  t a b l e s  o f  
Legendre-polynomials by G .  C .  Clark and S .  W .  Church i l l   and  
f r o m  t a b l e s  of s c a t t e r i n g  c o e f f i c i e n t s  made by C .  M .  Chu, 
G .  C .  Clark ,   and  S.  W .  C h u r c h i l l .  
T h i s  s t u d y  w a s  c a r r i e d  o u t  as par t  o f  a r e s e a r c h  program 
being performed under  NASA S u s t a i n i n g  G r a n t  NGR-43-001-021. 
* A s s i s t a n t  Professor of  Aerospace  Engineer ing .  
+ G r a d u a t e  R e s e a r c h  A s s i s t a n t .  
Introduction 
The  angular distribution of the  intensity of radiation 
scattered  at spherical particles was worked  out by G. Mie (1) 
in  theory. Several tables were prepared  for  particle  materials 
of real refractive index  (absorption by particle  material 
neglected).  In  these tables coefficients are given which have 
to  be combined with polynomials to give  the scattering dis- 
tribution function for single scattering at spherical particles 
of defined size in relation to the radiation wave  length. The 
scattering function is usually  for  an  incident beam of  unit 
intensity. 
This combination of coefficients and polynomials was 
done by a small computer program, based on the  data  provided 
by the tables for Legendre polynomials by G .  C. Clark  and S .  W. 
Churchill (2) and  by  the tables for scattering coefficients by 
C. M. Chu, G .  C. Clark and S. W. Churchill ( 3 ) .  
The computations became  necessary  because of an inten- 
tion  to  do a closer comparison of experimental results of 
scattering in  particle clouds with scattering predicted by 
the  Mie-theory  for spherical particles suspended in a gas. 
Mathematical Definitions and ComDutation 
As Hertel ( 4 )  proposed, the  normalized  scattering 
function f ( e )  may  be expressed by 
m 
f(P) = - ( 1 + c an - pn  (cos e >  
4T M=l 
2 
Here is f(e) = i ( 0 )  t h e   i n t e n s i t y  of a beam s c a t t e r e d   i n  
?rD2 K 
3- 
d i r e c t i o n  8 from a n  i n c i d e n t  beam of u n i t  i n t e n s i t y  h i t t i n g  a 
s p h e r i c a l  p a r t i c l e  
s c a t t e r e d  beam 
i n c i d e n t  beam o f  u n i t  of i n t e n s i t y  i (63) 
i n t e n s i t y  
s c a t t e r i n g  
p a r t i c l e  
The sca t te red  i n t e n s i t y  i ( 8 )  is independent  of cp for  s p h e r i c a l  
p a r t i c l e s  of uni form mater ia l  and a u n i f o r m  i r r a d i a t i o n .  
K is  t h e  s c a t t e r i n g  cross sec t ion  nond imens iona l i zed  
w i t h  t h e  g e o m e t r i c a l  cross s e c t i o n  D2 of t h e  s p h e r i c a l   p a r -  
t i c l e  of diameter D .  
77 
The s c a t t e r i n g  c o e f f i c i e n t s ,  an ,  are  independent   o f  t h e  
a n g l e ,  8 ,  b u t  v a r y  w i t h  p a r t i c l e  s ize  r e f e r r e d  t o  r a d i a t i o n  
wave l e n g t h  a = - i n  t h e  medium su r round ing  t h e  p a r t i c l e s  
and w i t h  t h e  p a r t i c l e  material  of r e f r a c t i o n  r e l a t i v e  t o  t h e  
su r round ing  medium. 
7r . D  
x 
The an were computed by a u t h o r s  (3) accord ing  t o  the  
r e l a t i o n s  d e r i v e d  by them. 
3 
where 
f 0  f o r  j # k 
'j k J J J 
= Re(A.)Re(Bk) + Im(Aj)Im(Bk) + Re(B.)Re(Ak) + Im(B.)Im(Ak), 
W j k n  = 0, if j + k - n 2 r ,  r = 0 , 1 , 2  ... k ,  
( L j + n - k ) ( L k + n - j ) ( ~ j + k - n ) ( L J + ~ + ~  2 l 2  
W j k n  - 
- 
i f j + k - n = 2 r , r = O ,  1, 2 . .   . k .  
vjkn = 0,  i f  j + k - n # 2 r  + 1, r = 0, 1, 2 . . . k ,  
and 
V (2n+l) ( j+k-n) (j+n-k+l) (k+n-j+l)(L j+n-k+l) ( L k+n-j+l)  ( L j+k-n-1) j k n  - - 
j+k+n+l ) 
( L 7  
where 
4(  L j+k+n+l) [ (L  j+n-k+l )   (Lk+n- j+l )  -"  (L j+k2n-1) 3 
i f  j +k - n = 2 r  + 1, r = 0 , 1 , 2 .  . . k  
I m  = Imag ina ry  pa r t ,  
Re = Real p a r t ,  
L = f a c t o r i a l  . 
The Legendre polynomials Pn(cos e )  were computed by a u t h o r s  
(2) up t o  t h e  h i g h  n e c e s s a r y  o r d e r  n by a r e c u r r e n c e  r e l a t i o n  
4 
! 
i 
For the  wanted scattering function 
W 
only  the sum of the products an - Pn has  to be  formed. 
Since the  input of the data an and Pn was the only time- 
consuming work, the  symmetry relation 
was used to necessitate  the  input of Pn for  the range of 
O' < 8 4 90° only  and  let  the  computer  fill  in  the Pn f o r  
90' 8 d - 180° by use of this symmetry relation. 
The f ( 0 )  were computed for  scattering angles 0 2 O ,  5O, 
loo, 15O, 20°, 25", . . . 160°, 165O, 170°, 175O, 17S0, adjusted 
to  the range of possible scattering measurements, the  particle 
size  parameter a took values a = 1, 2, 3 ,  4 ,  5, 6 ,  8, 10,  15, 
20, 25, 30, according to the  values  for which the coefficients 
a had  been computed in ( 3 ) .  The real index of refraction m 
covered  the range m = 1.20,  1.25, 1.30, 1.33, 1.44, 1.50,  1.55, 
1.60, 2.0, for  aerosols. 
n 
The  number of angles 8 turned o u t  to be chosen too small 
in  order to represent f(8) exactly  enough  for  higher a (15  to 30), 
because f ( 8 )  goes through  many  maxima  and  minima when scanning 
Oa ,< 8 5 180'. Since an experimental investigation by Love (5) 
has shown that f(e) probably cannot be  truly  represented  in  all 
maxima  and  minima  (no unique -a mixtures of particles  finite 
beams of radiation),  the calculations have  not been extended 
over  more intermediate 8 - values. 
5 
Tables  and Granhs 
T h e  c o m p u t e r  p r i n t o u t  g i v e s  4n . f ( @ )  v a l u e s  w r i t t e n  
i n  s e q u e n c e  a c c o r d i n g  t o  t h e  3 7  8 - v a l u e s 6  
The 4n f ( 8 )  - v a l u e s  a r e  g i v e n  t o  seven  decimal p l a c e s ;  how- 
e v e r  5 a re  o n l y  u s e f u l  b e c a u s e  of t h e  l i m i t a t i o n s  of t h e  t ab l e s  
( 2 ) ,   ( 3 ) .  The l a s t  t w o  p l a c e s  may be u s e d   f o r   r o u n d   o f f .  
A l i m i t e d  number o f  f ( 8 )  - v a l u e s  w a s  graphed t o  have 
a n  immediate p i c t u r e   o f   t h e   f u n c t i o n .  The o r d i n a t e  i n  t h e  g r a p h s  
is  s i g n e d  f ( e ) ,  a l t h o u g h  t h e  f a c t o r  4n is st i l l  c o n n e c t e d  t o  
f(Q> 
The g r a p h s  477 f ( 8 )  f o r  Q - v a l u e s  15,  20,   25,  30 do n o t  
t r u l y  r e p r e s e n t  the s c a t t e r i n g  f u n c t i o n  b e c a u s e  t h e  8 - steps 
are t o o  wide t o  g ive  enough  in fo rma t ion  ove r  t he  many maxima 
and  minima of f(e), w h i c h  o f t e n  e x t e n d  o v e r  a r a n g e  ne = 2 t o  5' 
o n l y .  
6 
Computer - Program for 
N 
4n f (0)  = 1 -+ 2 an pn  (cos e )  
n=l 
/INPUT 
/JOB GO 
DIMENSION P(70,37) ,A(70)  ,F(37) 
READ (5,111 ((P(1,J),I=1,70),J=1,19) 
11 FORMAT (10F8.0) 
Do 111 J=1, 18 
DO 111 1=1,69,2 
K=Jfl9 
L=- J+19 
P(I,K)=-P(1,L) 
111 P(I+l,K)=P(I+l,L) 
12 READ (5,13)N 
13  FORMAT (14) 
IF (N) 14 , 20 , 14 
10 FORMAT(7Fll. 0) 
1 5  F(J)=l .O 
14 READ  (5,lO)  (A(1) , I=l,N) 
DO 15 J=1,37 
DO 16 K=1,37 
DO 16 J=l,N 
WRITE  (6,17) F 
Go To 12 
END 
16 F(K)  =F(K)+A(J)*P(J,K) 
17 FORMAT  (1P5E16.7) 
20 STOP 
/DATA 
P(I,J) = Legendre  polynomials  Pn(cos 0)  for 37 0 - values-and 
orders  n  up to N = 70. 
i 
I 
I 
A (  I) = Scattering  coefficients  a n 
F(K) = 4 ~ f  (0) 
Refe rences  
1. Mie, G . ,  "Bei t rage   zur   Opt ik   t ruber   Medien" ,  Ann. Phys ik ,  
25,  377-445  (1906). 
2 .   C l a r k ,  G .  C . ,  S .  W .  Church i l l ,   "Tab les   o f   Legendre   Po ly -  
nomials" ,   Univ.   of   Michigan  Press ,   Engineer ing  Research 
I n s t .   P u b l i c a t i o n s ,   1 9 5 7 .  
3.  Chu, C .  M . ,  G .  C .  C l a r k ,  S .  W .  Churchi l l ,   "Angular  D i s t r i -  
b u t i o n  C o e f f i c i e n t s  f o r  R a d i a t i o n  S c a t t e r e d  by a S p h e r i c a l  
P a r t i c l e " ,   J o u r n .   O p t i c .   S O C .  America, 47,   (1957) ,  or 
i n  Un iv .  o f  Mich igan  P res s ,  Eng inee r ing  Resea rch  Ins t i t .  
P u b l i c a t i o n s .  
4 .  Hertel ,  W . ,  "Zur  Theor i e   de r   L ich t   s t r euung   du rch   t rube  
Schich ten"  Das Licht ,   40,   141-43,   165,   190-91,   214-15,  
232-34 , (1640) 
5.   Love, T .  J .  and R .  A .  Wheasler ,  "An Exper imenta l   S tudy   of  
I n f r a - R e d  S c a t t e r i n g  by Clouds of P a r t i c l e s " ,  A e r o s p a c e  
Resea rch  Labora to r i e s ,  ARL 64-109,  June  1964. 
8 
r 
"- 
! 
I I 
! I
i 
I 
I 
! 
~ 
i 
! 
! j m=1.25 
I 
4 
j ! 
I 
a=1 to 
9 
I 
I 
30 
20 
10 
8 
5 
3 
2 
t 
f ( e >  
1 .  
0.8 i. 
.- 
0 . 5  i 
c 
0.3 : 
0 . 2  , 
0.1 + 
0.08  - 
0 . 0 5  
I 
0.03 
I 
0.02 , 
0.01 i I I 
0 10 30 
I 
50 
Fig 
--- I 
I 
2 e o  
I 
I .  
I 
I "7 
' I  
10 
30 
2 0  
10 
8 
5 
3 
2 
f ( e >  
0.8 
0.5  
0.3 
0.2 
0.1 
0.08 
0.05 r 
" 
0.03 
0 . 0 2 .  
L .  
0 .0 lL  ~ .. 
0 :  
. .  
". 
. .  
. .  
m=1.33 
. ". 
i 
-t- 
~ 
! 
4 . .  
Fig. 3 
11 
3( 
2c 
1c 
E 
0.8 
0.5 
0.3 
0 . 2  
0.1 
0.08 
0.05 
0.03 
0.02 
0.01 
0 
! 
10 
! I  
- .  
. .  . . . . . . . 
. .  
. .  
50 70 
Fig. 4 
I 
1 I 
110  130 150 130 180 
12 
2 0  
10 
8 
5 
3 
2 
0.1 , 
0.08 . 
! 
0.05 , 
I 
I 
0.03 ; 
0.02 
0.01 I " - 1  
0 10 30 
, m=1.5 
! 
j I 
I 1 I 
50 70 
Fig. 5 
i 
i 
i 
i 
I 
I 
i 
90 
eo 
13 
. .  . . . . . . . .  
a = l  to 8 
. .  
! 
I  
! 
I 
i 
I 
110 
I 
I 
I 
f 
t 
I 
I 
I 
I 
i 
130  150
0 . 5  - . ~  
" 
0.3 . 
0.2  !" 
I 
j 
0.1 ~ 
0.08 + 
I 
L 
0.05 ; 
0.03 
0.02  - 
I 
I - 
0.01 " 
0 10 30 
I 
.I I 
m=1.55 a= 1 to 6 I 
. .  ." . . . .  - t  - "" - ... 
. ~. .. . . . . . . . . . . .  
1 I 
t 
I 
i 
I 
! 
i 
I 
I 
! 
I 
I 
i 
I 
I 
70 9c 
I 
i ; -  
~ 
I 
i 
, 
I 
I 
.. - - . " I 
1 1so 150 I 
2 
" 
.7 
. .  
0- -1 0 
Fig. 6 e" 
14 
f ( e >  
1 .  
0.8 + 
0 . 5  . 
0 . 3  . 
0.2 . 
0.1 , 
0.08 
0.05  . 
0.03 
0 . 0 2  
0.01 
0 
! 
I 
. .  
1 
10 
I 
m=1.6 
I 
I  I 
~ 
i 
! i 
I I 
50 70 
Fig. 7 
~ = l  to 8 
2 
90 110 13 0 150 170 I 
eo  
$0 
15 
30 
20 
10 
8 
5 
3 
0 .5  
0.3 
0 . 2  
0.1 
0.05  
0.03 
0 . 0 2  
0.01 
i 
I 
! 
1 ;  I I 
0 10 30 50 70 90 
Fig. 8 e o  
16 
1110 130 
I 
150 1'70 180 
NASA-Langley, 1968 - 24 CR-1022 
National Aeronautics and Space Administration 
WASHINGTON, D. C. 
FIRST CLASS. MAIL 
NATIONAL AERONAUTICS AN 
POSTAGE AND FEES PAID 
SPACE ADMINIS'IRAllON 
OFFICIAL BUSINESS 
POSTMASTER: If Undeliverable  (Section 1 5 :  
Postal Manual) Do Not Rem, 
"The aeronautical and space dctiuities of the United States shall be 
conducted so as to contribute . . . to the expansion of human knowl- 
edge of phenomena in the atmosphere and space. The Administrdion 
shall provide for the widest practicable and appropriate dissemination 
of information concerning its activities and the results tbereof." 
-NATIONAL AERONAUllCS AND SPACE ACT OF 1958 
NASA SCIENTIFIC AND TECHNICAL PUBLICATIONS 
TECHNICAL REPORTS:  Scientific and technical information considered 
important, complete, and a lasting contribution to existing.knowledge. 
TECHNICAL NOTES: Information less broad in scope but nevertheless of 
importance as a contribution to existing knowledge. 
TECHNICAL MEMORANDUMS: Information receiving limited distribu- 
tion because  of preliminary data, security  classification, or other reasons. 
CONTRACTOR REPORTS: Scientific and technical information generated 
under a NASA contract or  grant and considered an important contribution to 
existing knowledge. 
TECHNICAL TRANSLATIONS: Information published in a foreign 
language considered to merit NASA distribution in English. 
SPECIAL PUBLICATIONS: Information derived from or of value to NASA 
activities. Publications include conference proceedings, monographs, data 
compilations, handbooks, sourcebooks, and special bibliographies. 
TECHNOLOGY UTILIZATION PUBLICATIONS: Information on tech- 
nology used by NASA that may  be of particular intere;t in commercial and  other 
non-aerospace applications. Publications include Tech Briefs, Technology 
Utilization Reports and Notes, and Technology Surveys. 
Details on the  availability  of these publications  may  be  obtained from: 
SCIENTIFIC AND TECHNICAL INFORMATION DIVISION 
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
Washington, D.C. PO546 


Mie-scattering function

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server