ON THE MODIFICATION OF THE QUANTUM DEFECT METHOD APPLICABLE TO DENSE PLASMAS

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ON THE MODIFICATION OF THE QUANTUM DEFECT METHOD APPLICABLE TO DENSE

PLASMAS

A. Mihajlov, D. Djordjević, M. Popović

To cite this version:

A. Mihajlov, D. Djordjević, M. Popović. ON THE MODIFICATION OF THE QUANTUM DEFECT

METHOD APPLICABLE TO DENSE PLASMAS. Journal de Physique Colloques, 1979, 40 (C7),

pp.C7-689-C7-690. �10.1051/jphyscol:19797334�. �jpa-00219327�

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JOURNAL DE PHYSIQUE CoZZoque C7, s u p p l g m e n t au n 0 7 , Tome 40, J u i Z Z e t 1979, page C7- 689

ON THE MODlFICATICN OF THE QUANTUM DEFECT METHOC) APPLICABLE TO DMSE PLASMAS

A. Mihajlov, D. ~ j o r d j e v i g and M.M. popovic/.

I n s t i t u t e o f P h y s i c s , Beograd Y u g o s Z a v i a . I n a c a s e t h a t one u s e s t h e Bates and

~amgaard' method f o r bound-bound t r a n s i t i - ons and Burgess and s e a t o n 2 method f o r t h e bound-free t r a n s i t i o n s , it i s necessary t h a t t h e o u t e r e l e c t r o n and atom i n t e r a c - t i o n ( i n a s i n g l e e l e c t r o n approximation) h a s p u r e l y Coulomb behaviour f o r l a r g e r . I n c o n t r a s t t o t h i s c a s e under t h e plasma c o n d i t i o n s , t h e electron-atom i n t e r a c t i o n p o t e n t i a l f o r both t h e bound and f r e e e l e - c t r o n s t a t e s i s of a f i n i t e r a d i u s of a c t i - on (Debye p o t e n t i a l o r t r u n c a t e d Coulomb p o t e n t i a l ) and on account of t h a t t h e methods mentioned, when a p p l i e d t o plasma would g i v e r e s u l t s d i f f e r i n g from t h e ex- per iment

.

Our t a s k i s t o develop t h e method which, when a p p l i e d t o i s o l a t e d atoms auto- m a t i c a l l y g i v e s t h e r e s u l t s of t h e same accuracy a s Bates and Damgaard and Burgess and Seaton methods, w h i l e a t t h e same time it can be used under t h e c o n d i t i o n s charac- t e r i s t i c of plasma.

I t i s w e l l known t h a t t h e c o n d i t i o n r >>?o i s f u l f i l e d f o r a wide range of p l a s -

C

ma parameters

(Go

being a c o r e e f f e c t i v e r a d i u s and rc a s c r e e n i n g r a d i u s of t h e e f - f e c t i v e p o t e n t i a l ) , and t h i s i s f a c t which e n a b l e s u s t o n e g l e c t t h e i n f l u e n c e of a medium on behaviour of a system o u t e r e l e c t r o n + c o r e , i n a c a s e rsr,. This g i v e s

-

u s an o p p o r t u n i t y t o determine t h e parameters of t h e e f f e c t i v e p o t e n t i a l , t h a t would i n an a p p r o p r i a t e way d e s c r i b e t h e i n t e r a c t i o n of t h e system o u t e r ' e l e c t r o n + c o r e , by u s i n g t h e d a t a f o r an i s o l a t e d atom.

I n choosing t h e e f f e c t i v e p o t e n t i a l we w i l l assume, a s i s u s u a l , t h a t f o r

% > a o

t h e p o t e n t i a l i s a pure Coulomb: V= -l/r;

f o r %<ao we w i l l a d o p t t h e following po- t e n t i a l :

V ( r ) = - q / r ; q =

where a. i s t h e a n g u l a r quantum number of t h e ' s t a t e f o r which quantum d e f e c t i s

n e g l i g a b l e ; q r e p r e s e n t s t h e e f f e c t i v e charge which a c t s on t h e e l e c t r o n .

The b a s i c t a s k i n s o l v i n g t h e problem, reduces t o t h e d e t e r m i n a t i o n of ro and t h e dependence of t h e e f f e c t i v e c h a r g e q on t h e a n g u l a r quantum number R , and f o r a given 8 on t h e energy. The e f f e c t i v e r a d i u s i n our model depends o n l y on t h e c h o i c e of go, i n such a way t h a t ~ ( E , R ~ ) i s c l o s e t o u n i t y and i s of t h e o r d e r of magnitude of t h e c o r e r a d i u s Go; d e t e r m i n a t i o n of q i s li- mited by t h e c o n d i t i o n t h a t f o r t h e wave f u n c t i o n s corresponding t o t h e experimen- t a l l y determined energy of t h e s t a t e s t h e number of zero i s r e g u l a r .

A s i s u s u a l we w i l l t a k e r a d i a l wave f u n c t i o n R E r R ( r ) i n t h e form: R E P e ( r ) =

1 2

-P ( r )

.

For t h e bound s t a t e s ( E = -y /2) r E , R

f u n c t i o n P ( x ) (where x=2yr) s a t i s f i e s E;P-

t h e s t a n 3 a r d Wittacker e q u a t i o n , with para- m e t e r s e = q/y, u = R+1/2, and can be ex- pressed v i a i t s s o l u t i o n s M (x) ( f o r

X I u

r < r O ) and- ( x ) ( f o r r < r o ) . For t h e f r e e s t a t e s ( E = k /2) corresponding s o l u t i o n s '2 a r e : f o r r < r o

-

F R ( n , y ) and f o r r > r o

-

C ~ F ~ ( I I , Y ) + C 2 G R ( 7 - t I ~ ) where y=kr, n=q/k and a r e obtained from preceded s o l u t i o n s by u s i n g t h e formal substitution*+-irl, x+2iv.

-- ^A

I n t h e c a s e r=ro i t i s necessary t h a t t h e c o n d i t i o n of c o n t i n u i t y of t h e function and i t s d e r i v a t i v e a r e f u l f i l e d . For E < O t h e y a r e s a t i s f i e d w i t h t h e proper c h o i c e of q and t h e "sewing" c o n s t a n t ; f o r & < O

-

by c h o i c e of t h e c o e f f i c i e n t s C1 and

c2

w i t h q e x t r a p o l a t e d from t h e range E < O t o

& >O.

The phase which i s a consequence of t h e d e v i a t i o n of t h e e f f e c t i v e p o t e n t i a l from p u r e l y Coulomb one (--l/r f o r a11 r ) i s given by:

GR=arctg (C2/Cl) +jn (1)

where j i s a n a r b i t r a r y i n t e g e r .

The check of s e l f - c o n s i s t e n c y of t h e method when E+ZO ( t h e e q u a l i t y 6 & ( 0 ) =

r p ( 0 ) when j i n eq. (1) i s correspondingly R

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797334

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choosen), is performed by using the asymptotic expressions for the radial wave fun- ctions @, 7-19-, I)+-) and ~ ( 0 )

-

^{the value}

Of quantum defect extrapolated at E=O.

R e s u l t s :

The proposed method we tested in the case of the heavy noble gases; as an illustration we will present the results obtained for Argon.

Since in the case of Argon f states are practically hydrogenic, we adopted Q0=3. For ro we obtained $he value ro=1.7 which is consistent with the assumptions of the method (effective radius of the ~ r + ion is

Go

: 1,5).

On fig.1 we presented the dependence of the effective charge q on the angular quantum number

a

for ro=1.7 and for the main quantum number n=4 (which in the case of Argon corresponds to the first excited f state).

On fig.2 as an illustration, we give the 4 9 state wave functicn for 32r210; with crosses we displayed the behaviour of the wave function for small r, obtained by using the asymptotic expression for the P(%) function. It is obvious that the

Y I R

results of our calculation obtained for bound-bound transitions should be identical with ones obtained using the Bates and Dagmard method in the domain where it is applicable, if we use the same asymptotic normalization constant.

On fig.3 we presented the dependence q ( ~ ) for ~=0,1,2. On fig. 4 we give the dependence of the corresponding quantum defects u(s) for the same Q. By comparing figures 3 and 4 it becomes evident that q and u depends on E in the same way.

The phase & a which we obtained is in agreement with the corresponding,quantity 6 Q = ~ v Q (for j=2) that appears in the quantum defect method. In the range of energies in which the quantum defect method is applicable and 6% in a worst case, differ by some 5%.

As a consequence of the results presented it is obvious that the method proposed above, in the case of isolated atoms, automatically ensures an accuracy not

smaller than that of the Bates and Dagmaard method as well as Burgess and Seaton methcd, in a domain where they are applicable.

The advantage of the method we propose consist in the fact that, by using ones determined dependence q ( ~ ) for Qzao, it enables us to perform the calculations even under the conditions characteristic of dense plasmas when, for large r, V(r) dif- fers substantialy from the Coulomb poten- tial.

R e f e r e n c e s :

1. D.R.Bates,A.Damgaard,Phyl.Trans.A242, 101 (1949)

2. A.Burgess, M.J.Seaton, Mon.Not.Roy.

Astron.Soc.,

121,

76 (1960)

.

Fig. 3 e(E

S

. .-

^lo

Fig. 4