Physics of planetary atmospheres - Variation method used to calculate Rayleigh scattering cross sections of helium as wavelength functio

Chan, Y. N.

Dalgarno, A.

English

NASA Technical Reports Server

. 
GCA Technical Report No. 65-14-N 
PHYSICS OF PLANETARY ATMOSPHERES I: 
RAYLEIGH SCATTERING BY HELIUM 
Y. N. Chan 
A. Dalgarno 
GCA CORPORATION 
GCA TECHNOLOGY DIVISION 
Bedf ord, Massachusetts 
Contract No. NASW-1283 
Prepared f o r  
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
HEADQUARTERS 
Washington, D. C. 
April 1965 
https://ntrs.nasa.gov/search.jsp?R=19650021451 2020-03-17T01:07:03+00:00Z
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i .  
TABLE OF CONTENTS 
Title 
ABSTRACT 
INTRODUCTION 
THEORY 
CALCULATIONS 
REFERENCES 
Page 
1 
1 
3 
5 
10 
i 
PHYSICS OF PLANETARY ATMOSPHERES I: 
RAYLEIGH SCATTERING BY HELIUM 
By Y.M. Chan and A. Dalgarno t 
ABSTRACT 
3/05 22 
A var ia t iona l  method is used t o  ca lcu la te  the Rayleigh sca t t e r ing  
cross sect ions of helium as a function of wavelength. 
i s  3.53 x 10-26 cm2. 
The value a t  Lyman- 
INTRODUCTION 
The cross sect ion fo r  the scat ter ing of l i g h t  by an atomic gas may be 
obtained from the r e f r ac t ive  index. 
The r e f r ac t ive  index of helium has begn measured with high precision 
The 
a t  wavelengths between 27002 and 54002 [ l ]  and the measurements can be 
extrapolated to  longer wavelengths without ser ious loss  of accuracy. 
extrapolat ion t o  shor te r  wavelengths has been car r ied  out using a semi- 
empirical procedure [ 2 L b u t  the accuracy is  uncertain. 
can be calculated d i r e c t l y  a t  any wavelength by va r i a t iona l  procedures s i m -  
i l a r  t o  those employed i n  the calculation of s t a t i c  p o l a r i z a b i l i t i e s  and w i t h  
comparable accuracy. 
The r e f r ac t ive  index 
Our procedure, which r e l a t e s  the evaluation of i n f i n i t e  summations to  
the solut ions of d i f f e r e n t i a l  equations, d i f f e r s  i n  i t s  approach from those 
discussed by Karplus e t  a l .  [3-71. 
'Department of Applied Mathematics, The Queen's University of Belfast .  * 
Numbers i n  [ ] throughout the tex t  represent reference numbers. 
1 
THEORY 
The r e f r a c t i v e  index n of a gas of number densi ty  N i s  given by 
n - 1 = 2nNa(v) (1) 
where a ( v )  i s  the frequency-dependent atomic or  molecular po lar izabi l i ty .  
I f  t h e  atom or molecule i s  i n  a s ta te  with eigenfunction I#o and I#, represents  
the eigenfunctions of the other quantum s t a t e s  
v -v so 
where vso i s  the eigenfrequency of the t r a n s i t i o n  from $0 t o  
e l e c t r i c  dipole moment 
and m i s  the - 
m = 1 e.r - l-i 
i 
(3) 
ei being the charge of the i t h  p a r t i c l e  and r. being i t s  posi t ion vector. 
The p o l a r i z a b i l i t y  (2) can b F w r i t t e n  altern&vely a s  
o r  a s  
where ,m' i s  the e l e c t r i c  gradient  operator 
Now introducing a function X ( E . / v )  such t h a t  
1 
it  may be shown (cf. Dalgarno [8 ,9] )  t h a t  ( 4 )  i s  equivalent t o  
3 
and (5) to 
Equation (10) can be written alternatively as 
where 
(H - Eo + hv)x* ( v )  + g'q, 5 0 - 
Equation (7) m y  be solved by constructing a functional 
J(v) = {%(V) IH - Eo + hvlx<v)] + 2[X(VL *()] (13) 
and minimizing J(v) with respect to some trial form of ~(v). Then 
is a lower bound to the polarizability (provided JI 
of H). 
is an exact eigenfunction 0 
4 
CALCULATIONS 
W e  adopted f o r  the ground s t a t e  of helium the 20-parameter representa- 
t i o n  of Hart and Herzberg [ l o ]  and a t r i a l  funct ion X t ( v )  of the form 
the c (v) being va r i a t iona l  parameters. Subs t i tu  ed i n t o  (14) they y i e ld  a 
s t a t i c  p o l a r i z a b i l i t y  (v = 0) of 0.204 x 
the most accurate  value avai lable:  0.205 x c m  [ l l ] .  
S cm’ ig close agreement with 
As Table 1 demonstrates, the convergence of a ( v )  as a function of the 
number of va r i a t iona l  parameters i s  very rap id  f o r  the v i s i b l e  region of the 
spectrum but less so as the f i r s t  resonance wavelength is approached. The 
proper change i n  the s ign  of J (v)  does occur as the wavelength i s  decreased, 
the f i r s t  pole  occurring a t  5736: compared w i t h  the co r rec t  loca t ion  of 5842. 
Attempts t o  proceed t o  l a rge r  numbers of va r i a t iona l  parameters led t o  ser ious  
numerical i n s t a b i l i t i e s .  
The values of n - 1 measured by Cuthbertson and Cuthbertson El], which 
r e f e r  t o  wavelengths longer than 27502, exceed the predicted values by 0.44% 
near 90002, the excess increasing t o  0 . 5 6  a t  27502. 
increased the theoretical values uniformly by 0.47% and they are presented 
i n  Figures 1 and 2, which include also the  measured values.  
We have accordingly 
The cross  section f o r  the sca t t e r ing  of i s o t r o p i c  unpolarized r ad ia t ion  
i s  given by 
2 5 Q = -  12811 a ( v )  
3x4 
and values  of Q may be r ead i ly  derived from Figures 1 and 2. 
The cross  sec t ion  f o r  sca t te r ing  of Lyman-a i s  3.53 x 10-26 cm2. Values 
fo r  some other  wavelengths are given i n  Table 2. 
5 
TABLE 1 
Convergence of a(v)  
0 1 2 3 4 
~ ~ _ _ _  ~~~~~~ 
9110 1.132 1.378 1.379 1.379 - 1.379 
3037 1.147 1.413 1.414 1.414 1.414 
911 1.319 1.905 1.950 1.952 1.952 
759 1.422 2.307 2.452 2.465 2.467 
6 07 1.661 3.869 5.507 6.342 6.786 
569 1.775 5.288 13.18 56.86 - 38.61 
6 
3.65 
3.60 
10 
0 - 
x 3.55 
A - 
I 
t 
Y 
3.50 
3.45 
OICI23- 10: 
I I 1 
I I 1 
2000 6000 3000 4000 
X( i i ,  
5000 
Figure 1. The refractive index of helium. 
7 
I 
x -  
CI - 
I 
t 
Y 
E 
4 
3 
1 
I I 1 
2000 2500 3000 I 
Figure 2. The refractive index of helium. 
I 
8 
mLX 2 
L i g h t  Scattering Cross Sections Q cm 2 
~~~~ 
(2) 700 800 1000 1500 2000 2500 
9 
REFERENCES 
1. Cuthbertson, C .  and Cuthbertson, M., Proc. Roy. SOC. A135, 40 (1932). 
2. Dalgarno, A. and Kingston, A. E., Proc. Roy. SOC. A259, 424 (1960). 
3. Karplus, M., J. Chem. Phys. 37, 2773 (1962). 
4. Karplus, M. and Kolker, H. J., J. Chem. Phys. 39, 1493, 2997 (1963). 
5. Yaris, R . ,  J. Chem. Phys. - 39, 2474 (1963). 
-
6. Yaris, J. Chem. Phys. 40, 667 (1964). 
7 .  Levine, H. B. and Taylor, H. S., J.  Chem. Phys. 41, 1367 (1964). 
8. Dalgarno, A., Quantum Theory, Vol. 1, ed. D.R. Bates ( A c a d e m i c  Press ,  
New York, 1961). 
9. Dalgarno, A., Rev. Mod. Phys. 35, 522 (1963). 
10. Hart, A. and Herzberg, G . ,  Phys. Rev. 106, 79 (1957). 
11. Schwartz, C. ,  Phys. Rev. - 123, 1700 (1961). 
10 


Physics of planetary atmospheres. i- ray- leigh scattering by helium

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