A GENERAL CHARACTERISTIC EQUATION FOR A DIFFUSION-CONTROLLED POSITIVE COLUMN OF CIRCULAR CROSS SECTION WITH ONE-STEP AND TWO-STEP IONIZATION PROCESSES

HAL Id: jpa-00219492

https://hal.archives-ouvertes.fr/jpa-00219492

Submitted on 1 Jan 1979

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A GENERAL CHARACTERISTIC EQUATION FOR A DIFFUSION-CONTROLLED POSITIVE COLUMN OF CIRCULAR CROSS SECTION WITH ONE-STEP

AND TWO-STEP IONIZATION PROCESSES

L. Gerald Rogoff

To cite this version:

L. Gerald Rogoff. A GENERAL CHARACTERISTIC EQUATION FOR A DIFFUSION-

CONTROLLED POSITIVE COLUMN OF CIRCULAR CROSS SECTION WITH ONE-STEP AND

TWO-STEP IONIZATION PROCESSES. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-175-

C7-176. �10.1051/jphyscol:1979786�. �jpa-00219492�

JOURNAL DE PHYSIQUE Colloque C7, suppl6rnent au n07, Tome 40, JuiZZet 1979, page C7- 1-75

A GENERAL CHARACTERISTIC EQUATION FOR A DIFFUSION-CONTRaLED POSITIVE COLUMN OT CIRCULAR CROSS SECTION WITH ONE-STEP AND TWO-STEP IONIZATION PROCESSES

L. Gerald Rogoff.

Westinghouse Research and Development Center, Pittsburgh, Pennsylvania 15235 U.S.A.

A simple y e t g e n e r a l e q u a t i o n c h a r a c t e r i z e s form n e = n o g ( x / ~ , y / ~ ) , where t h e d e n s i t y no i s a t h e e l e c t r i c a l p r o p e r t i e s of a s t e a d y - s t a t e , c o n s t a n t r e p r e s e n t i n g t h e amplitude of t h e d i s t r i - l o n g i t u d i n a l l y - u n i f o r m p o s i t i v e column i n which t h e b u t i o n and t h e dimensionless f u n c t i o n g d e s c r i b e s e l e c t r o n d e n s i t y n i s g i v e n by t h e c o n t i n u i t y t h e s p a t i a l v a r i a t i o n i n terms of d i s t a n c e s normal- e q u a t i o n i z e d t o a s c a l e l e n g t h T t r a n s v e r s e t o t h e a x i s ,

2 i . e . , T i s an a r b i t r a r y r e f e r e n c e d i s t a n c e which

DV ne

+

vne

+

kne2 = 0 (1)

allows f o r a l i n e a r s c a l i n g of t h e s i z e of t h e w i t h t h e c o e f f i c i e n t s D , v, and k independent o f c r o s s s e c t i o n . (Rectangular c o o r d i n a t e s a r e used

* 2 p o s i t i o n and w i t h ne=O a s t h e boundary c o n d i t i o n . a r b i t r a r i l y f o r c l a r i t y . ) The o p e r a t o r VT

The g e n e r a l e x p r e s s i o n i s derived elsewhere1 f o r c o n t a i n s d e r i v a t i v e s w i t h r e s p e c t t o t h e normalized columns of a r b i t r a r y c r o s s - s e c t i o n a l shape (includ- d i s t a n c e s , and t h e normalized a r e a a i s r e l a t e d t o i n g i n t e r n a l s u r f a c e s n o t connected w i t h t h e o u t e r t h e a c t u a l a r e a A by aT 2 =A.

e n c l o s u r e ) w i t h t h e r a t e s l i n e a r and q u a d r a t i c i n For a d i s c h a r g e column d e s c r i b e d by Eq. ( l ) , n r e p r e s e n t i n g v a r i o u s p o s s i b l e e l e c t r o n produc- Eq. (2) i s a g e n e r a l e x p r e s s i o n r e l a t i n g t h e coef- t i o n and l o s s p r o c e s s e s ( i . e . , v and k p o s i t i v e o r f i c i e n t s , t h e number of e l e c t r o n s p e r u n i t l e n g t h . n e g a t i v e ; we assume D>O). That e x p r e s s i o n i s t h e c r o s s - s e c t i o n a l a r e a , and t h e shape of t h e

where A i s t h e c r o s s - s e c t i o n a l a r e a of t h e d i s - charge space, Ne i s t h e t o t a l number of e l e c t r o n s p e r u n i t l e n g t h of t h e column, and S i s a dimen- s i o n l e s s number c h a r a c t e r i s t i c of t h e shape of t h e column c r o s s s e c t i o n .

The q u a n t i t y S i s g i v e n by

where g i s a normalized e l e c t r o n d e n s i t y d i s t r i b u - t i o n , :V is arnormalized L a p l a c i a n , and d a i s a normalized element of c r o s s - s e c t i o n a l a r e a . That i s , t h e s o l u t i o n of Eq. (1) i s expressed i n t h e

c r o s s s e c t i o n , which i s involved i n t h e i n t e g r a l e x p r e s s i o n f o r S , Eq. ( 3 ) . S i n c e t h e c o e f f i c i e n t s a r e f u n c t i o n s of t h e a p p l i e d e l e c t r i c f i e l d , s i n c e Ne i s p r o p o r t i o n a l t o t h e t o t a l e l e c t r o n c u r r e n t , and s i n c e t h e e l e c t r o n c u r r e n t i s u s u a l l y approxi- mately e q u a l t o t h e t o t a l d i s c h a r g e c u r r e n t ,

Eq. (2) r e p r e s e n t s t h e v o l t a g e - c u r r e n t c h a r a c t e r i s - t i c of t h e column.

The i n t e g r a l i n Eq. ( 3 ) i s t o t a l l y normalized.

Thus, i f t h e form o f g i s f i x e d , t h e n t h i s i n t e g r a l i s independent of t h e s i z e of t h e c r o s s s e c t i o n , i.e., independent of T and A. It i s shown e l s e - where1 t h a t f o r any g i v e n combination of s i g n s of

v and k, t h e form of g i s f i x e d i f t h e r a t i o kno/v and t h e c r o s s - s e c t i o n a l shape a r e f i x e d . Thus, f o r

13

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979786

a g i v e n v a l u e o f kno/v w i t h t h e s i g n s o f v and k s p e c i f i e d , S is a d i m e n s i o n l e s s number c h a r a c t e r i s - t i c of t h e shape o f t h e column c r o s s s e c t i o n . Once S i s evaluated f o r given d i s c h a r g e c o n d i t i o n s ,

~ q . (2) r e p r e s e n t s a g e n e r a l i z e d c h a r a c t e r i s t i c f o r o t h e r d i s c h a r g e c o n d i t i o n s corresponding t o t h e same v a l u e of kn /v and same c r o s s - s e c t i o n a l shape.

Note t h a t t h e i n t e g r a l i n Eq. (3) i s w r i t t e n i n terms of normalized q u a n t i t i e s t o emphasize i t s independence o f c r o s s - s e c t i o n a l a r e a . S c a n a l s o b e w r i t t e n i n terms o f non-normalized q u a n t i t i e s a s

e l e c t r o n impact, by two-step e l e c t r o n impact, 3 by e x c i t e d state-ground s t a t e c o l l i s i o n s , and by e x c i t e d s t a t e - e x c i t e d s t a t e c o l l i s i o n s , a s w e l l a s e l e c t r o n l o s s by attachment. Likewise, t h e e f f e c - t i v e r a t e c o e f f i c i e n t k may i n c l u d e e f f e c t s of ion- i z a t i o n by two-step e l e c t r o n impact and by e x c i t e d s t a t e - e x c i t e d s t a t e c o l l i s i o n s , a s w e l l a s e l e c t r o n l o s s by e l e c t r o n - i o n volume recombination.

The r e l a t i o n s h i p s p r e s e n t e d h e r e a r e q u i t e g e n e r a l , and t h e y may a p p l y t o a v a r i e t y of t y p e s of d i s c h a r g e s . However, t h e y a t e p a r t i c u l a r l y ap- p l i c a b l e t o low-pressure nonequilibrium d i s c h a r g e s .

Curve 714902-A

We c o n s i d e r h e r e t h e s p e c i a l c a s e o f a c i r c u l a r c r o s s s e c t i o n w i t h b o t h v and k>O. (This

. .

corresponds,' f o r example, t o e l e c t r o n p r o d u c t i o n by one-step a n d / o r two-step electron-impact i o n i z a - t i o n . ) For t h i s c a s e we have e v a l u a t e d n u m e r i c a l l y t h e q u a n t i t y S , and we have o b t a i n e d a r e l a t i o n s h i p between S and kno/v f o r a l l p o s s i b l e v a l u e s of kno/v. The r e s u l t s a r e g i v e n i n Fig. 1, where f o r convenience t h e a b s c i s s a i s kno/(v+kno) =

(kno/v)/ [l+(kno/v)

1.

Thus, t h e l e f t - m o s t v a l u e corresponds t o a l i n e a r p r o d u c t i o n r a t e o n l y (k=O) i n Eq. (1) and t h e right-most v a l u e corresponds t o a q u a d r a t i c r a t e o n l y (v=O), w i t h a l l p o s s i b l e combinations o f t h e two i n between. Note t h a t t h e l e f t - m o s t v a l u e can b e o b t a i n e d a n a l y t i c a l l y 1 t o b e Snr(2.405) 2

.

I n t h i s c a s e Eq. (2) reduces1 t o t h e r e l a t i o n s h i p v / I P ( 2 . 4 0 5 / ~ ) o b t a i n e d by Schottky 2 s p e c i f i c a l l y f o r a c i r c u l a r c r o s s s e c t i o n o f r a d i u s R w i t h one-step e l e c t r o n - i m p a c t i o n i z a t i o n .

The terms i n Eq. (1) may d e s c r i b e v a r i o u s p r o c e s s e s . F o r example, t h e e f f e c t i v e $ i f f u s i o n c o e f f i c i e n t D may r e p r e s e n t f r e e o r ambipolar d i f f u s i o n . S i m i l a r l y , t h e e f f e c t i v e f r e q u e n c y v may i n c l u d e t h e e f f e c t s o f i o n i z a t i o n by one-step

kno v

+

^kno

Fig. 1

References 1. G.L. Rogoff, t o b e published.

2. W. Schottky, Phys. 2. 25, 635 (1924).

3. For t h e i o n i z a t i o n p r o c e s s e s i n v o l v i n g e x c i t e d p a r t i c l e s t o b e r e p r e s e n t e d by v o r k, t h e r e l e v a n t e x c i t e d p a r t i c l e d e n s i t i e s must, of course, v a r y a p p r o p r i a t e l y w i t h ne.