R A D I A T I V E R E C O M B I N A T I O N IN S O M E

I O N S OF A S T R O P H Y S I C A L I N T E R E S T

UDIT NARAIN and H. P. MITAL Astrophysics Research Group,

Physics Department, Meerut College, Meerut 250001, India

and

SURESH CHANDRA Physics Department,

D. N. Degree College, Gulaothi, Bulandshahar 245408, India

(Received 16 November; in revised form 30 December, 1976)

Abstract. Radiative recombination coefficients for some quadruply and quintuply ionized atoms, present in the Sun and its atmosphere, are investigated in the temperature range 10-104 K by using the method of detailed balance. Simple expressions are given for a quick estimation.

1. Introduction

The radiative recombination rates are of importance in many astrophysical problems such as, the investigation of ionization equilibrium of elements (House, 1964; Burgess and Seaton, 1964; Tucker and Gould, 1966; Jordan, 1969, 1970; Ansari et al., 1970; Landini and Monsignori Fossi, 1972; Jain, 1976), the temperature distribution deduced from intensity ratios (Waldmeier, 1952; Elwert, 1952; Schwartz and Zirin, 1959; Kang and Amy, 1968; Narain and Chandra, 1975, 1976; Chandra, 1976) in the solar corona, the study of thermal balance in gaseous nebulae (Seaton, 1958, 1959) and the spectrum of the solar corona (Seaton, 1964; Billings, 1966), etc.

Approximate total rate coefficients for radiative recombination can be com- puted by the formulae given by Elwert (1952) and Seaton (1959). For individual quantum states these coefficients may be evaluated accurately if the corresponding photoionization cross sections are accurately known (Bates and Dalgarno, 1962; Bates, 1974; Callaway, 1974; Chandra and Narain, 1976a). The method of Morse and Stueckelberg (1930) becomes extremely tedious for complex atoms.

Here we compute the rate coefficients for quadruply (0 +4, F +4, Ne +4, Na +4, Mg +4 and A1 +4) and quintuply (F +5, Ne +5, Na +5, Mg +5, A1 +5 and S +5) ionized

atoms using the method of detailed balance (Bates and Dalgarno, 1962).

2. Theoretical Details and Calculations

The rate coefficient for the process

X~m+~(n2~+aL)+ e --+ X+~(n2~'+IL')+ hu (1)

Solar Physics 52 (1977) 417-422. All Rights Reserved Copyright �9 1977 by D. Reidel Publishing Company, Dordrecht-Holland

418 U D I T N A R A I N E T AL.

TABLE I Values of the parameters

Target s C B a I (eV) ga gi

0+3 3 1/3 3.33 1.82 77.413 6 1 F +3 2 2/3 3.1 1 87.138 9 6 Ne +3 2 1 2.95 1.20 97.11 4 9 Na +3 2 4/3 2.7 1 98.91 9 4 Mg +3 2 5/3 2.3 1 109.31 6 9 A1 +3 2 2 1.9 1 119.99 1 6 F +4 2.3 1/3 2.4 1 114.24 6 1 N e +4 2.31 2/3 2.16 1 126.21 9 6 Na +4 2.3 1 2.0 1 138.40 4 9

Mg +'* 2.3 4/3 1.9 1 141.27 9 4 A1 § 2.3 5/3 1.8 1 153.75 6 9 Si § 2.3 2 1.6 1 166.77 1 6

TABLE II Radiative recombination coefficients (in cm 3 sec -1)

Temperature ( K ) 0 +4 F +4 Ne +4 N a +4 M g § A1 +4

10 1.620 -1~ 5.257 -11 2.758 -11 1.768 -1~ 6.812 -11 2.034 -11 20 1.146 -1~ 3.713 -11 1.950 11 1.250-1o 4.816-11 1.438-11 30 9.355 11 3.031 11 1.592-11 1.021-zo 3.933-11 1.174-11 50 7.246 -xl 2.348 -11 1.233 -11 7.905 -11 3.046 -11 9.097 -12 70 6.124 -11 1.985 -11 1.042 -11 6.681-11 2.575 -11 7.688 -12

100 5.124 -11 1.660 -11 8.722 -12 5.590 -11 2.154 -11 6.432 12 200 3.623 -11 1.174 -11 6.167 -a2 3.953 -11 1.523 -1I 4.548 -12 300 2.958 -11 9.586 -12 5.036 -12 3.227 11 1.244-11 3.714-12 500 2.292 -11 7.426 -12 3.901-12 2.500 -11 9.633 -12 2.877 -12 700 1.937 -11 6.276 -12 3.297 -12 2.113 -11 8.141 12 2.431-12

1000 1.620 -11 5.251-12 2.758 -I2 1.768 -11 6.812 -12 2.034 -12 2000 1.146 -11 3.713 -12 1.950 -12 1.250 11 4.816-1z 1.438-12 3000 9.355 -12 3.031-12 1.592 -12 1.021-11 3.933 -12 1.174 -12 5000 7.246 -12 2.348 -12 1.233 -12 7.905 -12 3.046 -12 9.096 13 7000 6.124 -12 1.985 -12 1.042 -12 6.681-12 2.574 -12 7.688 -13

10000 5.124 -12 1.660 -12 8.722 -13 5.590 -12 2.154 -12 6,432 -13

a The superscript denotes the power of 10 by which the number is to be multiplied.

is g i v e n b y t h e e x p r e s s i o n ( B a t e s a n d D a l g a r n o , 1 9 6 2 ) :

R ( i n c m 3 s -1) = (1/c2)(2/7r)l/2(mkT)-3/2-~i e x p (I/kT) •

x ](hv)EO(v) e x p ( - hv/kT) d ( h v ) (2)

I

~k.,-+ m + 1 in w h i c h ga a n d gi a r e t h e s t a t i s t i c a l w e i g h t s o f t h e i o n s X 2'~ a n d - - i ,

r e s p e c t i v e l y , I is t h e i o n i z a t i o n p o t e n t i a l o f Xa+m, T t h e t e m p e r a t u r e , k t h e

RADIATIVE RECOMBINATION IN SOME IONS

T A B L E I l l

R a d i a t i v e r e c o m b i n a t i o n coeff ic ients (in c m 3 sec -1)

4 1 9

T e m p e r a t u r e (K) F +5 N e +5 N a +s M g +5 A1 +s Si +5

10 5 . 3 9 5 -ga 2 . 9 6 8 -9 1 . 4 6 9 -9 9 . 7 8 2 -9 4 . 0 7 0 - 9 1 . 2 7 5 - 9

2 0 1 .907 -9 1 . 0 4 9 -9 5 . 1 9 5 -1~ 3 . 4 5 8 -9 1. 4 3 9 - 9 4 . 5 0 7 - 1 ~

3 0 1 . 0 3 8 -9 5 . 7 1 1 - 1 ~ 2 . 8 2 8 lO 1 . 8 8 3 - 9 7 . 8 3 2 - 1 o 2 . 4 5 3 - 1 o

5 0 4 . 8 2 5 - I ~ 2 . 6 5 4 -1~ 1 . 3 1 4 -1~ 8 . 7 4 9 lO 3 . 6 4 0 - 1 o 1 . 1 4 0 - 1 o

7 0 2 . 9 1 3 -1~ 1 . 6 0 2 -1~ 7 . 9 3 4 -11 5 . 2 8 2 -1~ 2 . 1 9 7 -1~ 6 . 8 8 3 - 1 1

100 1 . 7 0 6 -1~ 9 . 3 8 5 -11 4 . 6 4 7 -11 3 . 0 9 3 -1~ 1 .287 -1~ 4 . 0 3 1 - 1 1

2 0 0 6 . 1 0 5 -11 3 . 3 5 8 - I I 1 . 6 6 3 -11 1 .107 -1~ 4 . 6 0 6 -1~ 1 -443-11

3 0 0 3 . 5 6 3 11 1 . 9 5 9 -11 9 . 6 9 8 12 6 . 6 4 3 - 1 1 2 . 6 8 8 - 1 1 8 . 4 2 2 - 1 2

5 0 0 2 . 1 6 4 -11 1 .189 -11 5 . 8 8 5 12 3 . 9 2 8 11 1 . 6 3 3 - 1 a 5 . 1 1 8 12

7 0 0 1 . 7 2 2 11 9 . 4 5 9 - 1 2 4 . 6 8 2 - 1 2 3 . 1 2 7 - 1 1 1 . 3 0 0 -11 4 . 0 7 5 12

1 0 0 0 1 . 4 0 9 11 7 . 7 4 0 - 1 2 3 . 8 3 1 - 1 2 2 . 5 5 9 - 1 1 1 .064 -11 3 . 3 3 5 - 1 2

2 0 0 0 9 . 8 8 3 -12 5 . 4 2 8 -12 2 . 6 8 7 -12 1 .795 -11 7 - 4 5 9 -11 2 . 3 3 9 -12

3 0 0 0 8 . 0 6 4 -12 4 . 4 2 9 -12 2 . 1 9 2 -12 1 . 4 6 4 -11 6 -087 -12 1 .909 -12

5 0 0 0 6 . 2 4 2 -12 3 . 4 2 9 -12 1 . 6 9 7 -12 1 . 1 3 4 -11 4 - 7 1 3 -12 1 . 4 7 8 - 1 2

7 0 0 0 5 . 2 7 3 -12 2 . 8 9 7 -12 1 . 4 3 4 -12 9 . 5 7 9 -12 3 - 9 8 1 - 1 2 1 . 2 4 8 - 1 2

1 0 0 0 0 4 . 4 0 9 -12 2 . 4 2 2 -12 1 .199 -12 8 . 0 1 0 -12 3 . 3 2 9 -12 1 . 0 4 4 12

a T h e s u p e r s c r i p t d e n o t e s t he p o w e r of 10 b y w h i c h the n u m b e r is to be mul t ip l i ed .

T A B L E IV

T h e va lues of t he c o n s t a n t s in un i t s of c m 3 sec -1 d e g 1/z e V -2

T a r g e t K1 K2 K 3

0 +4 8 . 5 5 0 -14a __ __

F +4 2 . 1 8 7 -14 __ __

N e +4 9 . 2 4 9 -15 __ __

N a § 5 . 7 1 4 - I4 _ _

M g +4 1 .806 -14 _ _

AI +4 4 . 4 6 7 -15 - - __

F +5 - - 1 .307 -11 3 . 3 7 8 14

N e +5 __ 5 . 8 9 2 -12 1 . 5 2 1 - 1 4

N a +5 _ 2 . 4 2 6 -12 6 . 2 9 6 -15

M g +5 _ _ 1 . 5 5 0 - 1 1 4 . 0 1 3 - 1 4

A1 +5 - - 5 . 4 4 4 -12 1 . 4 0 8 -15

Si +5 - - 1 . 4 4 9 12 3 . 7 5 4 - 1 5

a T h e s u p e r s c r i p t d e n o t e s t he p o w e r of 10 b y w h i c h the n u m b e r

is to b e mul t ip l i ed .

B o l t z m a n n c o n s t a n t , h v t h e p h o t o n e n e r g y a n d Q(p) t h e p h o t o i o n i z a t i o n c r o s s -

s e c t i o n s f o r t h e i n v e r s e p r o c e s s

~ s + m + l / 2 s + l r ~ X+~m(n2S'+lL ') + hv A i ~n L ) + e. ( 3 )

T h e o t h e r s y m b o l s h a v e t h e i r u s u a l m e a n i n g s ( C h a n d r a a n d N a r a i n , 1 9 7 6 b ) .

420 U D I T N A R A I N E T AL.

IO.C

IO,~

Tu II.C

E U

t9 O--q i 1.5

I

I 2 . 0

Fig. 1.

I0 I 0 2 i0 3 i 0 4

T C K )

Radiative recombination rate coefficients for quadruply ionized atoms.

The photoionization cross-sections needed for the evaluation of the integral in Equation (2) for the ions under consideration are given by the expression (Seaton, 1958):

- / v\-s-ll O(v) = 10-18CB I~ ( --v-v ) ' + ( 1 - a ) t ~ o ) /cm2. (4)

1 \Po I

The values of the coefficients C, B, o~ and s are given in Table I (Seaton, 1958). The values of I are taken from Moore (1949).

The integral in Equation (2) when combined with Equation (4) was evaluated by numerical integration.

3. Results and Discussion

The radiative recombination rate coefficients computed using Equation (2) in the temperature range 10--10 4 K are displayed in Tables II and III. For the sake of comparison they have been exhibited in Figures 1 and 2 as a function of temperature. No other theoretical or experimental data seem to be available.

9.0

'~ I0 .0 I'q

oe

o o _.l I i 1 . 0

R A D I A T I V E R E C O M B I N A T I O N IN S O M E I O N S

12.0 00 10 2 IO 3 IO 4

T ( . K )

421

Fig. 2. Radiative recombinat ion rate coefficients for quintuply ionized atoms.

It is obvious from Figures 1 and 2 that the behaviour of the rate coefficients for all atoms in the same ionization state is similar. In a large temperature range they decrease linearly with the square root of the temperature, as may be expected in many cases of interest (Bates and Dalgarno, 1962).

In the case of quadruply ionized atoms the coefficients may be approximated by the expression

R1 = K I I 2 / T I/2, (5)

where I is in eV and T in K. The constants K1 for different ions are presented in Table IV. For quintuply ionized atoms,these rate coefficients may be represented by the following expressions:

R2 = K2rI2/T 3/2, T < 400 K (6)

and R3 = K312/T l/a, T > 400 K. (7)

These constants (K2 and K3) for different ions are also presented in Table IV. Since the accuracy of the rate coefficients strongly depends on the accuracy of

the photoionization cross-sections, we may expect our coefficients to be correct to within :t:20 per cent, i.e., the inaccuracy as given by Seaton (1958) for his photoionization cross-sections for positive ions.

422 t JDr r NARAIN ET AL.

Acknowledgements

W e are gra tefu l to P rofessor S. P. K h a r e for his e n c o u r a g e m e n t . O n e of us (S.C.)

is thankfu l to D r K. C. Mit ta l , the Pr incipal , D. N. D e g r e e Col lege , G u a l o t h i for

p rov id ing necessa ry facil i t ies and two of us (U.N. and H.P .M. ) a re thankfu l to the

U .G .C . , New De lh i and M e e r u t Un ive r s i ty for f inancial assis tance. Thanks are

also due to the re fe ree for his suggest ions in improv ing the manuscr ip t .

References

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