HAL Id: jpa-00223311
https://hal.archives-ouvertes.fr/jpa-00223311
Submitted on 1 Jan 1983
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.
CURRENT STATUS OF CALCULATIONS AND MEASUREMENTS OF ION STOPPING POWER IN
ICF PLASMAS
T. Mehlhorn, J. Peek, E. Mcguire, J. Olsen, F. Young
To cite this version:
T. Mehlhorn, J. Peek, E. Mcguire, J. Olsen, F. Young. CURRENT STATUS OF CALCULATIONS AND MEASUREMENTS OF ION STOPPING POWER IN ICF PLASMAS. Journal de Physique Colloques, 1983, 44 (C8), pp.C8-39-C8-66. �10.1051/jphyscol:1983804�. �jpa-00223311�
JOURNAL DE PHYSIQUE
Colloque C 8 , supplement au nO1l, T o m e 44, novembre 1983 page C8-39
CURRENT S T A T U S OF C A L C U L A T I O N S AND MEASUREMENTS OF I O N S T O P P I N G POWER I N I C F PLASMAS
T.A. Hehlhorn, J.M. Peek, E.J. McGuire, J.N. Olsen and F.C. youngs
Sandia NatioraZ Laboralorie;, ,llbuqv.erque, R . X . 8 7 1 9 5 , u.S.11.
*filavaZ Research I ~ b o r a t o r y , idaskington, D . C . , U . S . A .
RESUME
En fusion inertielle conduite par les ions, on 6prouve actuelle- ment la n6c6ssit6 de perfectionner les modeles de ralentisscment.
L16volution des recherches montre que les lois d16chelle approch6es ne sont plus suffisantes pour extrapoler les modsles actucllemcnt uti- lis6s. On se propose de predire, 3 10 % pres, les parcours ioniques dans les cibles ICF. On expose le modsle du gaz d161ectrons libres, ainsi que les profils de densit6 de charge atomique du type Hartree- Fock-Slater pour dgterminer l'ionisation moyenne I(Z,q,E) d'une cible d16lectrons. Cette rn6ttiode est syst6matiquement exp1ori.e afin de mettre en Gvidence les insuffisances de la physique sous-jaccnte, particulisrement pour de faibles vitesses des projectiles. Des modsles alternatifs sont 6galement d6velopp6s par d'autrcs auteurs 3 la Sandia.
Des mesures exp6rimentales de pouvoir dtarr6t, amplifi6 dans les plasmas de fusion, ont 6t6 observ6es dans lc domainc 0,3 T W / C ~ ~ au Naval Research Laboratory. Les exp6rimentateurs de la Sandia Ctendent actuellement ces donn6es 3 des 6tats d'ionisation plus 6lCvPe et 3 des cibles 3 Z plus grand, avec l'aide de l'accCl6rateur PROTO-1 ( 1 , 2 ~ ~ / c m * ) .
More p r e c i s e s t o p p i n g power models f o r u s e i n ICF t a r g e t d e s i g n need t o be developed. The l i g h t i o n beam ICF program i s now moving i n t o a phase where "ad h o c " s c a l i n g of c e r t a i n key p h y s i c s p a r a m e t e r s i n t h e s t o p p i n g power models i s no l o n g e r s u f f i c i e n t . Our g o a l i s t o p r e d i c t i o n r a n g e s i n ICF t a r g e t s t o w i t h i n a b o u t 10-205. A v e r i f i e d s t o p p i n g power model i s a l s o e s s e n t i a l i n d i a g n o s i n g t a r g e t
i r r a d i a t i o n i n t e n s i t i e s ; s u c h d a t a c a n o n l y be i n f e r r e d by t a r g e t r e s p o n s e . P r e s e n t l y , our a r e a of primary c o n c e r n i n v o l v e s
c a l c u l a t i n g t h e s t o p p i n g power of t h e bound e l e c t r o n s of p a r t i a l l y i o n i z e d atoms. One bound e l e c t r o n s t o p p i n g power model t h a t we a r e i n v e s t i g a t i n g u s e s t h e l o c a l o s c i l l a t o r model a l o n g w i t h E a r t r e e - F o c k - S l a t e r atomic c h a r g e d e n s i t y p r o f i l e s t o c a l c u l a t e I ( Z , q . E ) , a g e n e r a l i z e d a v e r a g e i o n i z a t i o n p o t e n t i a l f o r t h e t a r g e t e l e c t r o n s . T h i s method i s b e i n g s t u d i e d s y s t e m a t i c a l l y t o l o o k f o r d e f i c i e n c i e s i n t h e u n d e r l y i n g p h y s i c s model, e s p e c i a l l y a t low p r o j e c t i l e v e l o c i t i e s . Another p r o c e d u r e u s e s t h e G e n e r a l i z e d O s c i l l a t o r S t r e n g t h model t o c a l c u l a t e t h e bound e l e c t r o n s t o p p i n g .
E x p e r i m e n t a l measurements of enhanced s t o p p i n g power i n ICF p l a s m a s a t t h e 0.3 lW/cm l e v e l have been r e p o r t e d by t h e Naval Research L a b o r a t o r y . F u r t h e r e x p e r i m e n t s a t S a n d i a a r e aimed a t e x t e n d i n g t h i s d a t a b a s e b o t h t o h i g h e r i o n i z a t i o n s t a t e s and t o higher-Z t a r g e t s u s i n g a 1 . 2 lWIcm2 p r o t o n beam on t h e PROTO-I a c c e l e r a t o r .
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983804
JOURNAL DFI PHYSIQUE
The i m p l i c i t g o a l of o u r work o v e r t h e p a s t few y e a r s h a s been t o d e v e l o p t h e c o m p u t a t i o n a l a b i l i t y t o a c c u r a t e l y s i m u l a t e t h e s t o p p i n g power o f any a r b i t r a r y i o n i n a t a r g e t of a r b i t r a r y c o m p o s i t i o n , t e m p e r a t u r e , d e n s i t y , and d e g r e e of i o n i z a t i o n . We have r e c e n t l y made s i g n i f i c a n t p r o g r e s s towards t h i s goal; however, much work s t i l l r e m a i n s b e f o r e even t h e more modest n e e d s of S a n d i a ' s l i g h t i o n beam f u s i o n program a r e a d e q u a t e l y met.
T h i s workshop p r e s e n t s a good forum from which t o i s s u e a c a l l t o t h e i n t e r n a t i o n a l p h y s i c s community t o j o i n i n a n a t t e m p t t o d e v e l o p more a c c u r a t e and more g e n e r a l s t o p p i n g power models t o be u s e d i n i o n - d r i v e n ICF. The key h e r e i s t h e a c c u r a c y . Many u s e f u l models t h a t p r e d i c t g e n e r a l t r e n d s and t h e a p p r o x i m a t e s c a l i n g of s t o p p i n g powers f o r ICF p l a s m a s a l r e a d y e x i s t ; most of them a r e mentioned i n S e c t i o n 4. However, now i s t h e t i m e t o a t t e m p t t o make t h e s e models more a c c u r a t e by r i g o r o u s l y comparing them w i t h t h e l a r g e body of c o l d m a t t e r s t o p p i n g power d a t a and t h e n e x t e n d i n g them t o t h e more demanding regime of p a r t i a l l y i o n i z e d m a t t e r . S i n c e i t i s r e l a t i v e l y e a s y t o p e r f o r m good s t o p p i n g power e x p e r i m e n t s f o r c o l d t a r g e t s t h e r e h a s b e e n a t e n d e n c y t o t r e a t t h e i m p o r t a n t p h y s i c a l q u a n t i t i e s t h a t c o n t r o l t h e m a t e r i a l and p r o j e c t i l e d e p e n d e n c i e s of t h e s t o p p i n g p r o c e s s a s e m p i r i c a l p a r a m e t e r s t h a t a r e o b t a i n e d t h r o u g h b e s t f i t s t o e x p e r i m e n t a l d a t a . Good e x p e r i m e n t a l measurements o f i o n s t o p p i n g powers i n p a r t i a l l y i o n i z e d ICF p l a s m a s a r e n o t p a r t i c u l a r l y " e a s y " t o do.
Moreover, t h e d a t a w i l l p r o b a b l y n o t be o b t a i n e d w i t h a s h i g h a p r e c i s i o n a s c a n be a c h i e v e d w i t h c o l d t a r g e t s . T h e r e f o r e , t h e s t o p p i n g power models t h a t must be d e v e l o p e d f o r t h e d e s i g n of r e l i a b l e , and e f f i c i e n t ICF t a r g e t s w i l l need t o be more h i g h l y d e v e l o p e d and i n t e r n a l l y c o n s i s t e n t .
E x p e r i m e n t s a r e n e c e s s a r y t o v e r i f y t h e models, b u t t h e d a t a i s l i k e l y t o be t o o s p a r s e and t o o u n c e r t a i n t o r e l y on d e t e r m i n i n g e m p i r i c a l p a r a m e t e r s from t h e measurements. T h i s u n c e r t a i n t y i s a g g r a v a t e d by t h e f a c t t h a t f o r most e x p e r i m e n t s now e n v i s i o n e d one c a n o n l y v e r i f y t h e d e p o s i t i o n package i n c o n j u n c t i o n w i t h a hydrodynamics package; i t m i g h t be i m p o s s i b l e t o t e s t t h e s t o p p i n g power models s e p a r a t e l y . We h a v e t h e r e f o r e f o r m u l a t e d a n e i g h t p o i n t p l a n f o r d e v e l o p i n g a n a c c u r a t e and g e n e r a l s t o p p i n g power model f o r u s e i n ion-beam ICF t a r g e t d e s i g n and t a r g e t d i a g n o s t i c s :
1 ) Develop c a n d i d a t e s t o p p i n g power models f o r h y d r o g e n i c p r o j e c t i l e s ( e . g . p r o t o n s , d e u t e r o n s , t r i t o n s - h e r e a f t e r r e f e r r e d t o g e n e r i c a l l y a s p r o t o n s f o r s i m p l i c i t y ) e m p h a s i z i n g t h e a t o m i c p h y s i c s of t h e low t o
i n t e r m e d i a t e e n e r g y regimes. ( I n B e t h e f o r m a l i s m t h i s i s e q u i v a l e n t t o a c c u r a t e l y c a l c n l a t i n g t h e a v e r a g e i o n i z a t i o n p o t e n t i a l and t h e s h e l l c o r r e c t i o n s . )
2 ) T e s t t h e s e models a g a i n s t t h e l a r g e amount of c o l d t a r g e t d a t a f o r p r o t o n s t o p p i n g powers f o r e l e m e n t s a c r o s s t h e p e r i o d i c t a b l e .
3 ) Compare t h o s e models t h a t a r e s n c c e s s f u l i n S t e p 2 w i t h a v a i l a b l e p r o t o n s t o p p i n g power d a t a f o r p a r t i a l l y i o n i z e d t a r g e t s .
4 ) Develop a n e f f e c t i v e p r o j e c t i l e c h a r g e model f o r higher-Z i o n s t h a t i s c o m p a t i b l e w i t h t h e p r e v i o u s p r o t o n model and i s u s e f u l f o r a r b i t r a r y d e g r e e s of t a r g e t i o n i z a t i o n .
5 ) Compare t h i s combined model w i t h a c t u a l e x p e r i m e n t a l measurements of high-Z p r o j e c t i l e s t o p p i n g powers i n c o l d t a r g e t s .
6 ) Compare t h e combined model w i t h a v a i l a b l e s t o p p i n g power d a t a f o r p a r t i a l l y i o n i z e d t a r g e t s .
7 ) V e r i f y t h e d e n s e plasma s t o p p i n g power models a s t h e a v a i l a b l e i n t e n s i t y and e n e r g y on t a r g e t s i n c r e a s e s .
8 ) C o n t i n u e t o s t u d y whether plasma i n s t a b i l i t i e s a t h i g h i n t e n s i t i e s m i g h t l e a d t o anomalous s t o p p i n g powers o r o t h e r b e h a v i o r damaging t o o p t i m a l t a r g e t performance L1.21.
T h i s p r o g r e s s i o n i s l o g i c a l i f somewhat o b v i o u s , b u t i t i s meant t o p o i n t o u t t h a t a c a r e f u l l y c o n s t r u c t e d and v e r i f i e d model w i l l become i n c r e a s i n g l y i m p o r t a n t t o t h e a c c u r a t e c a l c u l a t i o n o f i g n i t i o n c o n d i t i o n s of i o n - d r i v e n ICF t a r g e t s . Note a l s o t h a t t h e s e s t e p s a r e n o t n e c e s s a r i l y i n d e p e n d e n t and s e q u e n t i a l . To q u a n t i f y o u r a c c u r a c y r e q u i r e m e n t s , w e e s t i m a t e t h a t e v e n t u a l l y we w i l l want t o be a b l e t o p r e d i c t i o n r a n g e s i n ICF t a r g e t m a t e r i a l s t o w i t h i n a b o u t 1 0 5 . I m p l i c i t i n t h i s agenda f o r d e v e l o p i n g a n a c c u r a t e t h e o r e t i c a l model f o r i o n s t o p p i n g power i n p a r t i a l l y i o n i z e d m a t e r i a l i s t h e need t o h a v e a p a r a l l e l e x p e r i m e n t a l e f f o r t t o p r o v i d e t h e d a t a n e c e s s a r y t o v e r i f y o r d i s p r o v e t h e models. T h i s i s c e r t a i n l y a n o n - t r i v i a l t a s k a n d i t would be u s e f u l t o s h a r e i d e a s a b o u t how t o o b t a i n such d a t a a t t h i s m e e t i n g .
I n c o n j u n c t i o n w i t h t h e i d e a of d e v e l o p i n g e x p e r i m e n t s f o r m e a s u r i n g on s t o p p i n g powers a t ICF-level beam i n t e n s i t i e s , we would emphasize t h e
C8-42 JOURNAL DE PHYSIQUE
i n t e r d e p e n d e n c e of t h e a b i l i t y t o r e l i a b l y p r e d i c t t h e s t o p p i n g power of an i o n beam and t h e a b i l i t y t o r e l i a b l y "measure" t h e beam i n t e n s i t y i n c i d e n t on a t a r g e t a t ICF o r near-ICF c o n d i t i o n s . That i s , a l l p r e s e n t t e c h n i q u e s f o r i n f e r r i n g t h e power and c u r r e n t d e n s i t y i n c i d e n t on a n ICF t a r g e t i n v o l v e coupled d e p o s i t i o n l h y d r o d y n a m i c s i m u l a t i o n of t h e t a r g e t i n t e r a c t i o n p r o c e s s . T h e r e f o r e , a v e r i f i e d d e p o s i t i o n l h y d r o d y n a m i c s i m u l a t i o n
c a p a b i l i t y i s e s s e n t i a l t o i n t e r p r e t t h e s e d i a g n o s t i c s c o r r e c t l y .
The n e x t s e c t i o n of t h i s p a p e r i s i n t r o d u c t o r y i n n a t u r e and g i v e s a n overview of t h e combined S a n d i a beam-target i n t e r a c t i o n model, of which t h e i o n s t o p p i n g power model i s o n l y a p a r t a l t h o u g h p e r h a p s t h e most
s i g n i f i c a n t one. The t h i r d s e c t i o n w i l l d i s c u s s some of t h e work b e i n g done a t S a n d i a on computing t h e c o n t r i b u t i o n of bound e l e c t r o n s t o t h e s t o p p i n g power of p r o t o n s i n p a r t i a l l y i o n i z e d plasmas. The n e x t s e c t i o n d i s c u s s e s
t h e p r e s e n t e x p e r i m e n t a l u n d e r s t a n d i n g of i o n s t o p p i n g powers under ICF c o n d i t i o n s . Both t h e p a s t work a t NRL a s w e l l a s e x p e r i m e n t s p r e s e n t l y underway a t S a n d i a w i l l be d e s c r i b e d . Moreover, t h e u n i q u e e x p e r i m e n t a l problems and c a l c u l a t i o n a l r e q u i r e m e n t s i n v o l v e d i n making such s t o p p i n g power measurements w i l l a l s o be d i s c u s s e d . The main p o i n t s of t h i s p a p e r w i l l be summarized i n t h e l a s t s e c t i o n .
2 BASIC SANDIA TABOET-INTBBACTION WDRL
A diagram showing t h e i m p o r t a n t components of t h e S a n d i a beam-target i n t e r a c t i o n model i s shown i n F i g u r e 1. The r a y t r a c k i n g module i n t e r f a c e s w i t h t h e hydrodynamic code and c o n t r o l s t h e s t o p p i n g power c a l c u l a t i o n s . T h i s module t r a c k s t h e i o n t r a j e c t o r i e s t h r o u g h t h e t a r g e t geometry, i n f o r m s t h e d e p o s i t i o n module of t h e p r o j e c t i l e and t a r g e t c o n d i t i o n s , and t h e n t r a n s l a t e s t h e c a l c u l a t e d AE d e p o s i t e d o v e r d i s t a n c e Ax i n t o a z o n a l energy
MODULE
I 4 SOLVE 7
F i g . 1. Block diagram of S a n d i a beam-target i n t e r a c t i o n model.
s o u r c e term f o r use i n t h e t e m p e r a t u r e e q n a t i o n of t h e hydro code.
Furthermore, beyond simply c o n t r o l l i n g t h e d e p o s i t i o n r o u t i n e s , t h i s module i s i n h e r e n t l y i m p o r t a n t s i n c e r e 1 i a b l e t a r g e t d e s i g n and s i m u l a t i o n a l s o r e q u i r e s t h e a b i l i t y t o a c c u r a t e l y model t h e s p a t i a l and angul a r dependence of t h e i n c i d e n t i o n beam. As d e p i c t e d i n F i g u r e 2, t h i s i o n " r a y t r a c k i n g "
i n a two-dimensional g r i d c a n e i t h e r occur a l o n g g r i d l i n e s (however, t h e s e g r i d l i n e s d i s t o r t a s t h e i m p l o s i o n p r o g r e s s e s so t h i s i s n o t a n a t t r a c t i v e o p t i o n ) , a l o n g 2-d r a y s a l l of which p a s s through t h e a x i s of symmetry, o r a l o n g 3-d r a y s whose a p p a r e n t p a t h s i n 2-d p r o j e c t i o n a r e h y p e r b o l i c .
SYMMETRY
F i g . 2 . P r o j e c t i o n of two- and three-dimensional r a y p a t h s t h r o u g h a c y l i n d r i c a l l y symmetric two-dimensional g r i d .
The d e p o s i t i o n c a l c u l a t i o n i s b a s e d on a n e f f i c i e n t and a c c u r a t e i n t e g r a t i o n o f a Bethe-type s t o p p i n g p o r e r e q u a t i o n . We e x p e c t t h a t t h i s b a s i c n u m e r i c a l frame-work w i l l remain unchanged i n t h e f u t u r e . R a t h e r , more s o p h i s t i c a t e d p h y s i c s w i l l be i n c l u d e d a s new s c a l i n g p a r a m e t e r s i n t h e Bethe-based model. For example, a s w i l l be d i s c u s s e d l a t e r , we have computed a new s e t of e f f e c t i v e a v e r a g e i o n i z a t i o n p o t e n t i a l s < I > f o r allrminnm u s i n g t h e modified-LOM model 131. These new v a l u e s were t h e n i n c o r p o r a t e d i n t o t h e a p p r o p r i a t e p h y s i c s s u b r o u t i n e , b u t t h e c a l c u l a t i o n a l a s p e c t s of t h e o v e r a l l model remained unchanged. New plasma s t o p p i n g power p h y s i c s o r a l t e r n a t e e f f e c t i v e c h a r g e models f o r heavy i o n s c o u l d l i k e w i s e be i n c o r p o r a t e d i n t h e a p p r o p r i a t e p h y s i c s s u b r o u t i n e s w i t h o u t a f f e c t i n g t h e numerical a l g o r i t h m s . T h e r e f o r e , by u s i n g t h i s e f f e c t i v e parameter
t e c h n i q u e t h e speed and a c c u r a c y of t h e numerical a l g o r i t h m s c a n remain o p t i m i z e d w h i l e i n c r e a s i n g l y more s o p h i s t i c a t e d p h y s i c s i s s i m u l a t e d .
T h i s f i n a l module i n our beam-target i n t e r a c t i o n model c a l c u l a t e s
JOURNAL D€ PHYSIQUE
i n f l i g h t r e a c t i o n r a t e s and y i e l d s . N u c l e a r and a t o m i c p r o c e s s e s t h a t g e n e r a t e d e t e c t a b l e r a d i a t i o n y i e l d s c a n e i t h e r be u s e d t o p r e d i c t c u r r e n t d e n s i t y on t a r g e t o r be u s e d t o i n f e r a c t u a l s t o p p i n g power i n f o r m a t i o n under ICF c o n d i t i o n s . The primary a t o m i c d i a g n o s t i c u s e d i n i o n - d r i v e n ICF i s measurement of c h a r a c t e r i s t i c x-ray e m i s s i o n due t o d i r e c t i n n e r - s h e l l i o n i z a t i o n c a u s e d by t h e i n c i d e n t c h a r g e d p a r t i c l e s (41. T h i s t e c h n i q u e . which h a s b e e n t h e o r e t i c a l l y i n v e s t i g a t e d by N a r d i and Zinamon 151, h a s t h e a d v a n t a g e t h a t i t c a n be imaged t o p r o v i d e s p a t i a l i n f o r m a t i o n a b o u t t h e d e p o s i t i o n r e g i o n . However, i t s use i s l i m i t e d t o low i n t e n s i t i e s because i o n i z a t i o n of t h e t a r g e t c h a n g e s t h e b a s i c c r o s s s e c t i o n s . Moreover, a f a i r l y s o p h i s t i c a t e d t r e a t m e n t of t h e vacancy p r o d u c t i o n c r o s s s e c t i o n s , f l u o r e s c e n t y i e l d s [61, and s u b s e q u e n t t r a n s p o r t of t h e l i n e - r a d i a t i o n t h r o u g h t h e t a r g e t m a t e r i a l a s a f u n c t i o n of d e n s i t y , t e m p e r a t u r e , and i o n i z a t i o n s t a t e i s r e q u i r e d t o f u l l y u t i l i z e t h i s t e c h n i q u e . Such models a r e b e i n g s y n t h e s i z e d from t h e a v a i l a b l e a t o m i c p h y s i c s d a t a b a s e (e.g.
r e c e n t work a t S a n d i a on p r e d i c t i n g t h e o b s e r v e d s p e c t r a from M-shell i o n i z a t i o n of g o l d [71) and w i l l c o n t i n u e t o be of d i a g n o s t i c importance.
D e t e c t i o n of c h a r a c t e r i s t i c r a d i a t i o n from n u c l e a r r e a c t i o n s h a s b e e n f r e q u e n t l y u s e d i n d i a g n o s i n g beam c o n d i t i o n s i n ICF L8.91. F i g u r e 3 summarizes t h e r e a c t i o n r a t e e q u a t i o n and a l s o c a t a l o g s some of t h e n u c l e a r r e a c t i o n s t h a t have been s u c c e s s f u l l y u t i l i z e d f o r ICF d i a g n o s t i c s . The model e q u a t i o n p o i n t s o u t t h e i n t e r d e p e n d e n c e of t h e beam c u r r e n t and t h e
s t o p p i n g power model. The c o n s t a n t "C" i n f r o n t of t h e i n t e g r a l c o n t a i n s t h e i n s t a n t a n e o u s v a l u e of t h e i n c i d e n t c u r r e n t ; t h u s , t h e r e a c t i o n r a t e , P, i s p r o p o r t i o n a l t o t h e i n s t a n t a n e o u s c u r r e n t and i n v e r s e l y p r o p o r t i o n a l t o t h e s t o p p i n g power. T h e r e f o r e , a c c u r a t e knowledge of one of t h e s e
q u a n t i t i e s p e r m i t s c a l c u l a t i o n of t h e o t h e r . F i g u r e 4 i l l u s t r a t e s two c r o s s s e c t i o n s of i n t e r e s t t o ICF. Note t h a t t h e energy dependence of d i f f e r e n t c r o s s s e c t i o n s c a n v a r y so a s t o e f f e c t i v e l y change t h e d e t e c t o r r e s p o n s e . For example, t h e v e r y narrcnr Breit-Wigner r e s o n a n c e i n t h e l i t h i u m c r o s s s e c t i o n makes t h i s m a t e r i a l a v e r y s e n s i t i v e d e t e c t o r of p r o t o n s slowing- down t h r o u g h t h e 440 keV peak, b u t h a s v i r t u a l l y no r e s p o n s e t o lower energy p r o t o n s .
I n c e r t a i n s p e c i a l c a s e s we might need t o become concerned w i t h o t h e r p r o c e s s e s such a s &-ray t r a n s p o r t , knock-on p r o d u c t i o n and t r a n s p o r t , and t h e l i k e . For example, i n a s e r i e s of D-T f u e l e d e x p l o d i n g p u s h e r t a r g e t s s h o t on t h e PROTO-I a c c e l e r a t o r , we c o n f i r m e d t h e s u s p i c i o n [l01 t h a t t h e main n e u t r o n y i e l d d i d n o t come from t h e r m o n u c l e a r r e a c t i o n s b u t r a t h e r from
t h e beam f u s i o n r e a c t i o n s of knock-on d e u t e r o n s and t r i t o n s [ I l l . These knock-ons where produced by l a r g e - a n g l e R u t h e r f o r d and n u c l e a r s c a t t e r i n g of
t h e p r o t o n beam by t h e f u e l i o n s . A s p e c i a l s e t of r o u t i n e s t o s i m u l a t e t h i s p r o c e s s w e r e w r i t t e n and i n c o r p o r a t e d i n t o our g e n e r a l i n t e r a c t i o n model. An example of t h e agreement found between experiment and c a l c u l a t i o n f o r t h i s p r o c e s s i s found i n F i g u r e 5 .
F i g . 3 . Smnmary of i n f l i g h t r e a c t i o n model e q u a t i o n . Also shown a r e some of t h e n u c l e a r r e a c t i o n s t h a t have been u s e d i n i o n - d r i v e n ICF d i a g n o s t i c s .
20 100 l000
Energy (KeV)
?O ' L ~ ( ~ . Y ) " B ~ I j
- b 1
h,
1
100 1000
Energy (KeV)
Fig. 4. Sample energy dependent c r o s s s e c t i o n s f o r ( d , t ) f u s i o n r e a c t i o n and ( L i , p ) gamma p r o d u c i n g r e a c t i o n .
JOURNAL DE PHYSIQUE
I I I I
X-RAY PULSE 14 MEV NEUTRONS
0.0 0.4 0.8 1.2 1.6 2 .O
TIME (IO-'SEC)
F i g . 5. Comparison of d e t e c t o r r e s p o n s e and c a l c u l a t i o n f o r p r o d u c t i o n of 1 4 MeV n e u t r o n s from e x p l o d i n g p u s h e r t a r g e t ( P r o t o - I Shot
# 3121).
The b a s i c p r o t o n s t o p p i n g power e q u a t i o n s [ l 2 1 u s e d i n t h e S a n d i a beam- t a r g e t i n t e r a c t i o n model c a n be summarized a s :
where:
a n d N i s Avogadro's number, p i s t h e t a r g e t d e n s i t y , A , Z and q a r e t h e 0
a t o m i c w e i g h t , n m b e r and i o n i z a t i o n s t a t e of t h e t a r g e t atom, < I > i s t h e a v e r a g e i o n i z a t i o n p o t e n t i a l , Ci i s t h e s h e l l c o r r e c t i o n f o r t h e ith s h e l l , ye i s t h e r a t i o of t h e i o n v e l o c i t y t o t h e e l e c t r o n t h e r m a l v e l o c i t y , w i s
P t h e plasma f r e q u e n c y , and t h e r e m a i n i n g symbols have t h e i r u s u a l
d e f i n i t i o n s . The u s e of t h e B e t h e e q u a t i o n f o r t h e bound e l e c t r o n s and t h e u s e of a s e p a r a t e term f o r t h e plasma e l e c t r o n s i s common b o t h t o t h i s work
and t o t h a t of Nardi, P e l e g and Zinamon 1131. k s h e r 1141. B a n g e r t e r 1151, and Beynon 1161. We p r e s e n t l y assume t h a t o u r dense plasma, f r e e e l e c t r o n s t o p p i n g power models a r e a d e q u a t e and t h a t our major u n c e r t a i n t y i s i n o a r modeling of t h e bound e l e c t r o n s t o p p i n g power of p a r t i a l l y i o n i z e d atoms.
P r e s e n t e x p e r i m e n t a l d a t a s u g g e s t s t h a t t h i s i s a v a l i d assumption. W i t h i n t h e B e t h e e q u a t i o n formalism, a n u n c e r t a i n t y i n t h e bound e l e c t r o n s t o p p i n g power f o r p r o t o n s c a n be d i r e c t l y r e l a t e d t o p h y s i c s u n c e r t a i n t i e s i n o u r c a l c u l a t i o n of t h e a v e r a g e i o n i z a t i o n p o t e n t i a l . < I > , and i n t h e r e l a t e d s h e l l - c o r r e c t i o n terms, C.. Once a g a i n , our r e s t r i c t i o n h e r e t o p r o t o n s i s t o a l l o w u s t o i g n o r e e f f e c t i v e c h a r g e e f f e c t s and t o o n l y s t u d y t h e v a r i a t i o n o f <I> and Ci w i t h i o n i z a t i o n .
A b r u t e f o r c e c a l c u l a t i o n of <I> i s s t i l l p r o h i b i t i v e l y d i f f i c u l t f o r a r b i t r a r y i o n s ; t h e formal d e f i n i t i o n of < I > i n t h e B e t h e t h e o r y b e i n g :
where E a r e a l l p o s s i b l e p o s i t i v e energy t r a n s i t i o n s of t h e t a r g e t atom,
n f n
a r e t h e c o r r e s p o n d i n g d i p o l e o s c i l l a t o r s t r e n g t h s , and (Z-q) i s t h e number of e l e c t r o n s . For ICF we need < I ( Z , q ) > f o r b o t h n e u t r a l s and f o r a l l i o n s of i n t e r e s t . To c o m p l i c a t e m a t t e r s f u r t h e r , i o n e n e r g i e s of a few MeVIamn g i v e o p t i m a l beam-target c o n p l i n g . I n t h i s energy regime, s h e l l c o r r e c t i o n s and t h e d e t a i l s of t h e low energy s t o p p i n g power p l a y an e s s e n t i a l r o l e i n energy d e p o s i t i o n and r a n g e c a l c n l a t i o n s . T h e r e f o r e , what we r e a l l y need f o r a11 atoms and t h e i r a s s o c i a t e d i o n s i s b o t h an average i o n i z a t i o n p o t e n t i a l and a s e t of energy-dependent s h e l l c o r r e c t i o n s . A l t e r n a t i v e l y , we can g e n e r a l i z e t h e d e f i n i t i o n of t h e a v e r a g e i o n i z a t i o n p o t e n t i a l such t h a t i t i s now energy dependent: I ( Z , q , E ) , where E i s t h e i o n energy. (The normal d e f i n i t i o n t h e n t a k e s t h e form I ( Z , q ) = I ( Z , q , E - > W ) ) . Thus, o u r g o a l of d e v e l o p i n g a n a c c u r a t e atomic s t o p p i n g power model h a s now become t h e g o a l of a c c u r a t e l y c a l c u l a t i n g t h e f u n c t i o n I (Z, q, E).
I n many of t h e a r t i c l e s c i t e d above, some b a s i c s c a l i n g f o r m u l a f o r I ( Z , q ) i s proposed b a s e d e i t h e r on f i t s t o approximate c a l c u l a t i o n s o r on simple, a n a l y t i c , p h y s i c a l models. For example, o u r o r i g i n a l s c a l i n g e s t i m a t e s t o o k t h e form I121 :
i n our p r e s e n t n o t a t i o n . T h i s f o r m u l a i s a c c u r a t e i n two l i m i t s , q=O and q=Z-l, t h e n e u t r a l atom and h y d r o g e n i c limits. Such v a l u e s , I ( Z , q ) , a t t e m p t t o approximate t h e h i g h energy l i m i t of t h e a v e r a g e i o n i z a t i o n p o t e n t i a l and
CA-48 JOURNAL DE PHYSIQUE
t h e y c a n be u s e f u l i n e l e c t r o n t r a n s p o r t o r approximate h i g h energy i o n beam t r a n s p o r t c a l c u l a t i o n s . However, an a s s o c i a t e d s e t of energy-dependent s h e l l c o r r e c t i o n s a r e r e q u i r e d f o r l i g h t i o n f u s i o n s i m u l a t i o n . Moreover, e x p e r i m e n t s show t h a t such s c a l i n g i s a d e q u a t e o n l y f o r hydrocarbon
compounds and o t h e r low-Z t a r g e t s . T h e r e f o r e , more a c c u r a t e models f o r c a l c u l a t i n g I ( Z . q ) a r e needed.
The Local O s c i l l a t o r Model of L i n d h a r d and Winther [ l 7 1 h a s b e e n f r e q u e n t l y u s e d f o r d e t e r m i n i n g t r e n d s i n I ( Z ) f o r n e u t r a l s [181, f o r c a l c u l a t i n g a p p r o x i m a t e s h a p e s t o t h e s t o p p i n g power c u r v e [191, e t c . T h i s method h a s a l s o b e e n c a l l e d t h e F r e e E l e c t r o n Gas Model, t h e L o c a l D e n s i t y Approximation, and t h e l i k e . These names a r e s u g g e s t i v e of t h e f a c t t h a t t h i s model t r e a t s t h e r a d i a l e l e c t r o n c h a r g e d e n s i t y about a n atom a s a l o c a l l y uniform, d e g e n e r a t e f r e e e l e c t r o n g a s w i t h a unique l o c a l plasma f r e q u e n c y . The s t o p p i n g power of a f a s t i o n i s t h e n g i v e n by t h e
p o l a r i z a t i o n d r a g o f t h e e l e c t r o n g a s on t h e i o n a s i t p a s s e s t h r o u g h t h e atom. W i t h i n t h e L i n d h a r d f o r m a l i s m t h e a v e r a g e i o n i z a t i o n p o t e n t i a l i s d e f i n e d a s [ l 7 1
where p ( r ) i s t h e s p h e r i c a l l y a v e r a g e d e l e c t r o n c h a r g e d e n s i t y of t h e atom o r i o n , w i s t h e e l e c t r o n plasma f r e q n e n c y . and y i s a n e m p i r i c a l c o n s t a n t
P
t h a t a p p r o x i m a t e s e l e c t r o n b i n d i n g e f f e c t s and h a s a nominal v a l u e of &.
The a t t r a c t i v e n e s s of t h i s approach i s t h a t one a p p a r e n t l y need o n l y
c a l c u l a t e t h e r a d i a l d e n s i t y p r o f i l e p ( r ) t o s u c c e s s f u l l y a p p l y t h i s method.
However, i t h a s b e e n h a s r e c e n t l y e s t a b l i s h e d t h a t t h e LOM model f o r t h e a v e r a g e i o n i z a t i o n p o t e n t i a l d o e s n o t s c a l e p r o p e r l y t o t h e h y d r o g e n i c l i m i t 1201. That i s , I(Z,Z-l) s c a l e s a s z ' ' ~ I ( I , o ) r a t h e r t h a n a s ~ ' 1 ( 1 , 0 ) a s i t s h o u l d . Thus t h e LOM model, which does a r e a s o n a b l e j o b of c a l c u l a t i n g a v e r a g e i o n i z a t i o n p o t e n t i a l s and s t o p p i n g powers f o r n e u t r a l atoms, w i l l be p r o g r e s s i v e l y more i n e r r o r a s t h e i o n i z a t i o n s t a t e of a n i o n i s i n c r e a s e d .
An a t t e m p t h a s been made t o augment t h e LOM t h e o r y by g e n e r a l i z i n g t h e e m p i r i c a l c o n s t a n t y i n t o a s i m p l e f u n c t i o n of q and Z t h a t would g i v e t h e c o r r e c t o n e - e l e c t r o n l i m i t [211. I n d e v e l o p i n g t h i s f u n c t i o n , p u b l i s h e d GOS ( G e n e r a l i z e d O s c i l l a t o r S t r e n g t h ) r e s u l t s f o r alnminum i o n s 1221 were used a s a benchmark. The GOS model i s b a s e d on t h e p l a n e wave Born a p p r o x i m a t i o n (FWBA). The s t r e n g t h of t h i s method i s t h a t i t i s r e l a t i v e l y s i m p l e t o u s e , p e r m i t s t h e c a l c u l a t i o n of a l l r e l e v a n t p r o c e s s e s , and i s v a l i d o v e r a l a r g e
e n e r g y range [231. For p r o t o n s t o p p i n g power c a l c u l a t i o n s i t s major drawback i s t h a t t h e PWBA method i s n o t v a l i d n e a r t h e e l e c t r o n i o n i z a t i o n
and e x c i t a t i o n t h r e s h o l d s . T h i s t h r e s h o l d r o u g h l y c o r r e s p o n d s t o p r o t o n e n e r g i e s of between 1 0 0 and 200 keV. Below t h i s energy t h e method i s n o t v a l i d , and n e a r t h e t h r e s h o l d t h e method t e n d s t o o v e r - p r e d i c t t h e s t o p p i n g power c r o s s s e c t i o n s . The GOS method h a s b e e n s u c c e s s f u l l y compared w i t h e l e c t r o n impact 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 b o t h n e u t r a l and i o n i z e d t a r g e t s ( i n c l u d i n g g o l d i o n s ) and i s c o n t i n u a l l y b e i n g compared t o
e x p e r i m e n t a l d a t a f o r o t h e r p r o c e s s e s a s i t becomes a v a i l a b l e . T h i s s e t of I - v a l u e s , c a l c u l a t e d u s i n g t h e GOS model, almost c e r t a i n l y r e p r e s e n t s t h e most a c c u r a t e v a l u e s t h a t have b e e n p u b l i s h e d t o d a t e . Using t h i s d a t a ,
good agreement was o b t a i n e d between t h e LOM and GOS I - v a l u e s f o r t h e alrminum s e r i e s of i o n s when t h e c o n s t a n t y was g e n e r a l i z e d t o t h e f u n c t i o n [ZOl :
F i g u r e 6 d e m o n s t r a t e s t h e agreement o b t a i n e d between t h e LOM and GOS models when t h i s g e n e r a l i z e d f u n c t i o n i s u s e d t o c a l c u l a t e t h e s t o p p i n g power. The o r i g i n a l model a l o n g w i t h t h i s g e n e r a l i z e d d e f i n i t i o n of 7 h a s been termed t h e augmented-LOM (A-MM) model. As w i l l be d i s c u s s e d f u r t h e r i n t h e n e x t s e c t i o n , t h e s e augmented-LOMIGOS-equivalent s t o p p i n g powers seem t o
c o r r e s p o n d v e r y w e l l w i t h t h e enhanced d e u t e r o n s t o p p i n g powers measured i n t h e ~ 1plasma of t h e + ~ NRL s t o p p i n g power experiment C241.
Energy k W
F i g . 6 , Comparison of p r o t o n s t o p p i n g power i n aluminum a s a f u n c t i o n of e n e r g y : Augmented-MM model ( s o l i d l i n e ) , GOS model ( d i a m o n d s ) , Z i e g l e r a n a l y t i c f i t t o expgriment ( d o t t e d l i p ? ) . P l o t A f o r +,,
n e u t r a l a l m i n u m , B f o r A1 plasma, C f o r A 1 , and D f o r A1 plasma.
JOURNAL DE PHYSIQUE
We nor seemingly have a f a i r l y r e l i a b l e s e t of a s y m p t o t i c I - v a l u e s f o r t h e i o n i c s e r i e s of a l m i n u m ; b u t what a b o u t t h e g e n e r a l i z e d v a l u e s I(Z.q,E) t h a t we need t o compute a c c u r a t e s t o p p i n g powers f o r e n e r g i e s of about 5 Mev/amu and below? The p r o c e d u r e t h a t r e f o l l o w e d t o g e n e r a t e t h e v a l u e s I ( 1 3 , q , E ) u s i n g t h e A-UIM method can be summarized a s f o l l o w s . F i r s t we g e n e r a t e d t h e s t o p p i n g number p e r e l e c t r o n , L, u s i n g t h e A-UIM model a s g i v e n by t h e e q u a t i o n :
The r a d i a l c h a r g e d e n s i t y p ( r ) was c a l c u l a t e d u s i n g f r e e atom w a v e f u n c t i o n s o b t a i n e d from a s t a n d a r d Herman-Skillman code l251; ground s t a t e o r b i t a l s were u s e d t h r o u g h o u t . The f u n c t i o n c ( p , E ) i s t h e Lindhard-Winther s t o p p i n g f u n c t i o n f o r a z e r o t e m p e r a t u r e Fermi d e g e n e r a t e e l e c t r o n g a s ( y = l ) . Approximations t o t h i s f u n c t i o n have been t a b u l a t e d and p u b l i s h e d by many a u t h o r s [17,191. Having v e r i f i e d t h e s e v a l u e s a g a i n s t t h e GOS r e s u l t s ( a s d i s c u s s e d a b o v e ) , t h e A-MM s t o p p i n g n m b e r s were t h e n e q u a t e d t o t h e formal d e f i n i t i o n of t h e L-value i n B e t h e t h e o r y , and a s e r i e s of I-values.
I ( 1 3 , q . E ) r e r e computed. These I - v a l u e s were t h e n f i t t o polynomials f o r u s e i n t h e s t a n d a r d beam-target i n t e r a c t i o n package d e s c r i b e d i n S e c t i o n 3 . The v a l u e s 1 ( 1 3 , q , E ) g e n e r a t e d by t h i s p r o c e d u r e a r e shown i n F i g u r e 7.
Proton Energy W V )
F i g . 7. V a r i a t i o n of t h e a v e r a g e i o n i z a t i o n p o t e n t i a l f o r aluminmu i o n s a s o f u n c t i o n ogl9nergy. Lowest c u r v e f o r n e u t r a l atom, h i g h e s t c u r v e f o r A 1 ion.
The p r o t o n r a n g e s i n a l u m i n m p r e d i c t e d by t h e s e I - v a l u e s a r e s u b s t a n t i a l l y d i f f e r e n t from t h o s e o b t a i n e d w i t h o u r o r i g i n a l s c a l i n g f o r m u l a (Eq. 3 ) . F i g u r e 8 s h o r s t h e r a n g e of a 1 . 6 MeV p r o t o n i n a n aluminum plasma a s a f u n c t i o n of t a r g e t i o n i z a t i o n . (The t a r g e t i o n i z a t i o n was o b t a i n e d v i a a Saha e q u a t i o n by assuming a c o n s t a n t d e n s i t y o f . O 1 t i m e s
s o l i d and t h e n v a r y i n g t h e t e m p e r a t u r e . ) The most s t r i k i n g f e a t u r e of t h i s p l o t i s t h a t t h e p r o t o n r a n g e h a s a p l a t e a u a t q=3. T h i s f e a t u r e i s due t o t h e c l o s e d - s h e l l c o n f i g u r a t i o n of t h e neon-like ~ lion. + ~We f i n d t h a t on a s u b s h e l l b a s i s t h e s t o p p i n g power of t h e t i g h t l y bound neon-like c o r e
changes v e r y l i t t l e between i o n i z a t i o n s t a t e s q=2 and q=3. Likewise, t h e 3 s e l e c t r o n i n t h e sodium-like c o n f i g u r a t i o n i s so weakly bound t h a t i t i s v i r t u a l l y i n d i s t i n g u i s h a b l e from t h e f r e e plasma e l e c t r o n s i n i t s s t o p p i n g power. T h e r e f o r e , b o t h t h e t o t a l s t o p p i n g power and t h e r e b y t h e range of a 1.6 MeV p r o t o n changes v e r y l i t t l e between c h a r g e s t a t e s q=2 and q=3 i n a n aluminum plasma. The maximum d e v i a t i o n between t h e LOH/GOS r e s u l t and t h a t g i v e n by Eq. 3 o c c u r s f o r q=3. F o r t n i t o n s l y , t h i s i s t h e i o n i z a t i o n s t a t e t h a t was r e a c h e d a t peak power i n t h e NRL s t o p p i n g power e x p e r i m e n t s . Thus, i t was r e l a t i v e l y e a s y t o s e e t h a t t h e r e s u l t s w e r e much c l o s e r t o t h e s t o p p i n g powers measured i n t h e e x p e r i m e n t s t h a n t h o s e c a l o a l a t e d u s i n g Eq. 3.
l0
Scaled-Bethe Model
0 - .-.-.----. LOMIGOS Model -
- -
6 - -
4 . . .
0 1 2 3 4 6 ~ 7 8 O l O l l P
Ionization State
F i g . 8. Range of 1.6 MeV p r o t o n i n a l o l i n u m a s f u n c t i o n of t h e d e g r e e of i o n i z a t i o n of t h e t a r g e t . S c a l e d B e t h e model (Eq. 3 ) and LOMIGOS r e s u l t s s h o r n . Note t h e marked d i f f e r e n c e i n t h e two models f o r ZBAR=3.
We have t r i e d t h i s g e n e r a l i z e d 7 f u n o t i o n f o r a g o l d t a r g e t t o s e e i f t h e same good agreement e x i s t s between t h e LOM and t h e GOS models f o r high-Z atoms. U n f o r t u n a t e l y , t h e agreement was n o t n e a r l y a s good a s f o r aluminum
JOURNAL D€ PHYSIQUE
atoms, e s p e c i a l l y a t low e n e r g i e s ( s e e F i g u r e 9 ) . T h i s s u g g e s t s t h a t t h e y f u n c t i o n i s n o t u n i v e r s a l . F u r t h e r m o r e , even though we c o u l d p o s s i b l y g e n e r a t e a n o t h e r unique f u n c t i o n f o r g o l d , t h i s i s n o t l e a d i n g t o t h e g e n e r a l i t y t h a t we a r e s e e k i n g . Moreover, i t i s p a r t i c u l a r l y d i s t r e s s i n g t h a t even t h e n e u t r a l atom r e s u l t s p r o v e t o be a r e l a t i v e l y bad f i t t o b o t h t h e GOS and t h e t a b u l a t e d e x p e r i m e n t a l r e s u l t s .
Energy MeV)
F i g . 9 . Comparison of p r o t o n s t o p p i n g power i n g o l d a s a f u n c t i o n of e n e r g y : Augmented-MM model ( s o l i d l i n e ) , GOS model (diamonds), Z i e g l e r a n a l y t i c f i t t ~ ~ g x p e r i m e n t ( d o t t e d I d n e ) . P l o t A f g s , n e u t r a l gold, B f o r An plasma, C f o r ~ l + , and D f o r A1 plasma.
A t p r e s e n t , it seems u s e f u l t o s u b j e c t t h e LOM model t o f u r t h e r s t u d y and a t t e m p t t o i s o l a t e any p h y s i c s d e f i c i e n c i e s i n i t s approach. T h e r e f o r e , r e a r e b e g i n n i n g t o s y s t e m a t i c a l l y s t u d y t h e v a r i o u s a s p e c t s o f , and
a p p r o x i m a t i o n s made i n t h e s e c a l c u l a t i o n s . T h i s means f i r s t r e t u r n i n g t o a s t u d y of n e u t r a l atoms t o f a c i l i t a t e comparison w i t h e x p e r i m e n t a l d a t a . The f i r s t s t e p i n t h i s p r o c e s s i s t o i n v e s t i g a t e t h e s e n s i t i v i t y of t h e
c a l c u l a t i o n s t o t h e e x a c t d e t a i l s of t h e c h a r g e d i s t r i b u t i o n , F i g u r e 10 d e m o n s t r a t e s t h e d i f f e r e n c e between t h e c a l c u l a t e d c h a r g e d i s t r i b u t i o n f o r a n i s o l a t e d atom ( d e n o t e d HFS f o r H a r t t e e - F o c k - S l a t e r ) and t h a t of t h e same atom i n a s o l i d - s t a t e l a t t i c e a s t a b u l a t e d i n Ref. 26. F i g u r e 11 shows t h e r e l a t e d change i n t h e c a l c u l a t e d p r o t o n s t o p p i n g power a s compared t o e x p e r i m e n t u s i n g t h e s e two d i s t r i b u t i o n s . (Note t h a t t h e i n t e r s t i t i a l
e l e c t r o n s n o t i n c l u d e d i n t h e t a b u l a t e d s o l i d - s t a t e d i s t r i b u t i o n a r e i n c l u d e d a s d e s c r i b e d i n Ref. 2 7 ) . We s e e t h a t w h i l e t h e low energy s t o p p i n g power i s o v e r - p r e d i c t e d f o r i s o l a t e d atoms, i t i s someahat under- p r e d i c t e d when t h e more r e a l i s t i c s o l i d - s t a t e d i s t r i b u t i o n i s used. (We a l s o s e e t h e o v e r - p r e d i c t i o n of t h e GOS model n e a r t h r e s h o l d a s p r e v i o u s l y n o t e d . ) As a n u m e r i c a l c h e c k of o u r p r o c e d u r e s we have compared o u r I - v a l u e s w i t h t h o s e r e p o r t e d by Z i e g l e r [191. For n e u t r a l atoms w i t h y = l o u r n u m e r i c a l p r o c e d u r e s s h o u l d be e q u i v a l e n t a s s u b s t a n t i a t e d by t h e good agreement shown i n T a b l e I. Note t h a t u s i n g t h e c a n o n i c a l v a l u e y=J& t h e c a l c u l a t e d I - v a l u e s f o r t h e two d i s t r i b u t i o n s a r e : HFS - 1 2 4 eV, MTW -
1 3 8 eV. T h i s i s t o be compared w i t h t h e a c c e p t e d e x p e r i m e n t a l v a l u e of 1 6 2 eV. Thus, t h e s o l i d s t a t e d i s t r i b u t i o n d o e s g i v e a b e t t e r e s t i m a t e t o I ( 1 3 . 0 ) . b u t a s u b s t a n t i a l d i s c r e p a n c y s t i l l e x i s t s . Our hope i s t h a t by u s i n g t h e b e s t a v a i l a b l e d a t a f o r p ( r ) f o r v a r i o u s n e u t r a l atoms ( i n c l u d i n g high-Z atoms s u c h a s g o l d ) we c a n i s o l a t e and c o r r e c t any i n a c c u r a c i e s o r o m i s s i o n s i n t h e i n t e r a c t i o n f u n c t i o n I ( p , E ) . We a r e c u r r e n t l y p r e p a r i n g t o perform s i m i l a r c o m p a r i s o n s between i s o l a t e d and s o l i d - s t a t e d i s t r i b u t i o n s f o r copper (2=29) and s i l v e r (2=47). We w i l l a t t e m p t t o e s t a b l i s h w h e t h e r a t r e n d t o w a r d s i n c r e a s i n g e r r o r s w i t h i n c r e a s i n g Z e x i s t s , and t o q u a n t i f y how much of t h e s e d e v i a t i o n s a r e due t o i n a c c u r a c i e s i n t h e c h a r g e d i s t r i b u t i o n .
F i g . 1 0 . R a d i a l c h a r g e d e n s i t y p r o f i l e s f o r n e u t r a l aluminurn atoms.
R e s u l t s f o r f r e e atoms (=F-S) and atoms w i t h i n a s o l i d - s t a t e l a t t i c e (M-J-W) a r e shown. I n t e r s t i t i a l s i n c l u d e d i n s o l i d - s t a t e d i s t r i b u t i o n u s i n g c o n t i n u o u s o p t i o n a s d i s c u s s e d i n Ref. 2 6 .
JOURNAL DE PHYSIQUE
Energy Mev)
F i g . 11. Comparison of p r o t o n s t o p p i n g power i n n e u t r a l almuinum a s a f u n c t i o n of energy: Free-atom LOM-calculation ( s o l i d l i n e ) , W S r e s u l t s (diamonds), e x p e r i m e n t a l d a t a ( d a s h e d l i n e ) , and s o l i d - s t a t e L O E c a l c u l a t i o n (chain-dot l i n e ) . N o t i c e t h e change i n t h e LOM-computed c u r v e s w i t h d i f f e r e n t c h a r g e d i s t r i b u t i o n s .
TABLE I
Comparison of a v e r a g e i o n i z a t i o n p o t e n t i a l s c a l c u l a t e d i n t h i s s t u d y w i t h t h o s e r e p o r t e d by Z i e g l e r [191. E-F-S d e n o t e s H a r t r e e F o c k - S l a t e r e l e c t r o n d i s t r i b u t i o n s f o r f r e e atoms. M-J-W d e n o t e s s o l i d - s t a t e
d i s t r i b u t i o n s f o r a t o m i c e l e c t r o n s a s t a b u l a t e d i n Ref. 26. The c o n t i n u o u s and d i s c o n t i n u o u s sub-headings r e f e r t o t h e way t h a t t h e i n t e r s t i t i a l e l e c t r o n s n o t i n c l u d e d i n t h e s o l i d - s t a t e t a b u l a t i o n s a r e j o i n e d t o t h e s o l i d - s t a t e d i s t r i b u t i o n .
M-J-W
U-F-S D i s c o a t i n a o u s C o n t i n w u s
T h i s Stndy 106.8 118.9 119.2
Z i e g l e r 106.3 118.7 ---
At t h i s p o i n t , a p a r t i a l l i s t of q u e s t i o n s r e l a t e d t o t h e LOM model and i t s l o n g term a p p l i c a b i l i t y f o r o u r modeling c a n be compiled:
1 ) What i s t h e most a p p r o p r i a t e p ( r ) t o use i n t h e s e c a l c u l a t i o n s f o r n e u t r a l atoms?
a ) Thomas Fermi b ) Hartree-Fock C ) S o l i d - s t a t e
2 ) What i s t h e most a p p r o p r i a t e p ( r ) f o r i o n s ?
3 ) W i l l a r e l a t i v i s t i c a t o m i c s t r u c t u r e code (e.g. Liberman [281) g i v e b e t t e r r e s u l t s f o r high-Z atoms and i o n s s u c h a s g o l d ? (Sample c a l c u l a t i o n s s u g g e s t t h a t t h i s i s o n l y a 10-2M e f f e c t [291).
4 ) I s t h e low energy regime p r o p e r l y t r e a t e d w i t h i n t h e Lindhard-Winther i n t e r a c t i o n model?
a ) What a b o u t c h a r g e exchange energy l o s s ? 1301 b ) Momentum t r a n s f e r ?
c ) Does t h e e f f e c t i v e c h a r g e of a p r o t o n e n t e r i n ?
d ) What i s t h e magnitude of More's o r b i t c o r r e c t i o n [31J?
5) Do h i g h e r o r d e r Z-proj e c t i l e e f f e c t s p l a y a r o l e i n t h i s modeling?
6) What impact d o e s t h e f i n i t e - t e m p e r a t u r e e x t e n s i o n of t h e L i n d h a r d i n t e r a c t i o n model of Maynard and Deutsch L321 have on t h e s e problems?
P e r h a p s some of t h e s e q u e s t i o n s c a n be a t l e a s t p a r t i a l l y answered a t t h i s workshop.
The f i n a l r e s u l t of i n t e r e s t i n t h e a r e a of bound e l e c t r o n s t o p p i n g powers f o r i o n i z e d m a t t e r i s t o d i r e c t l y s t u d y t h e GOS r e s u l t s f o r g o l d
133.341. We h a v e i n c o r p o r a t e d t h e t a b u l a t e d GOS v a l u e s i n t o o u r d e p o s i t i o n model. Low energy v a l u e s a r e i n t e r p o l a t e d u s i n g a s q u a r e r o o t of e n e r g y s c a l i n g .
F i g u r e 1 2 d e m o n s t r a t e s t h e v a r i a t i o n of t h e r a n g e i n g o l d w i t h i o n i z a t i o n f o r p r o t o n s of v a r i o u s e n e r g i e s . GOS r e s u l t s and t h o s e u s i n g Eq. 3 s c a l i n g a r e shoan. As i n t h e c a s e of aluminum, t h e r a n g e i s p r e d i c t e d t o d e c r e a s e more s l o w l y w i t h i o n i z a t i o n when t h e b e t t e r atomic p h y s i c s (GOS) i s used. The new and s u r p r i s i n g r e s u l t i s t h a t t h i s model p r e d i c t s t h a t t h e range f o r h i g h energy p r o j e c t i l e s 0 5 MeV/amu) c a n i n i t i a l l y i n c r e a s e w i t h i o n i z a t i o n of t h e atom. T h i s r e s u l t seems t o have been a n t i c i p a t e d by Brueckner [351, a l t h o u g h t h e mechanism h e i n v o k e s f o r t h i s anomaly d o e s n o t
seem t o be v a l i d h e r e . Such range l e n g t h e n i n g seems t o o c c u r o n l y f o r high-Z m a t e r i a l s , and we b e l i e v e t h a t t h i s r e s u l t c a n be d e s c r i b e d i n t e r m s of t h e d i f f e r e n c e s i n t h e i n t e r a c t i o n v e l o c i t i e s of t h e t a r g e t e l e c t r o n s when t h e y a r e i n t h e i r bound and t h e i r f r e e s t a t e s . When t h e e l e c t r o n i s bound, i t s c h a r a c t e r i s t i c v e l o c i t y i s d e t e r m i n e d by quantum s t a t i s t i c s and i s e s s e n t i a l l y g i v e n by t h e l o c a l Fermi v e l o c i t y . Looking a t a A U + ~ plasma, where t h e e f f e c t i s maximized, we f i n d t h a t t h e plasma t e m p e r a t u r e
