AB INITIO CALCULATIONS OF THE CHARGE STATE OF A FAST HEAVY ION STOPPING IN A FINITE TEMPERATURE TARGET

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Submitted on 1 Jan 1983

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AB INITIO CALCULATIONS OF THE CHARGE STATE OF A FAST HEAVY ION STOPPING IN A

FINITE TEMPERATURE TARGET

D. Bailey, Y. Lee, R. More

To cite this version:

D. Bailey, Y. Lee, R. More. AB INITIO CALCULATIONS OF THE CHARGE STATE OF A FAST HEAVY ION STOPPING IN A FINITE TEMPERATURE TARGET. Journal de Physique Colloques, 1983, 44 (C8), pp.C8-149-C8-158. �10.1051/jphyscol:1983810�. �jpa-00223317�

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Colloque C8, supplement au nO1l, Tome 44, novembre 1983

AB I N I T I O CALCULATIONS OF THE CHARGE STATE OF A FAST HEAVY I O N STOPPING I N A F I N I T E TEMPERATURE TARGET'

D. B a i l e y , Y.T. Lee and R.M. More

U n i v e r s i t y of C a l i f o r n i a , Lawrence Livermore National Laboratory,

P. 0 . Box 5508, L-477, Livermore, CA 94550, U.S.A.

RESUME

On presente un calcul de l ' e t a t de charge dependant du temps d'un ion lourd t r a v e r s a n t une c i b l e a temperature f i n i e . Les equations d ' e v o l u t i o n s o n t i n t e g r e e s ti l ' a i d e d ' u n modele d'atome moyen.

ABSTRACT

We present a c a l c u l a t i o n of t h e time dependent charge s t a t e of a heavy p r o j e c t i l e t r a v e r s i n g a f i n i t e temperature t a r g e t . The c a l c u l a t i o n uses an average-atom model t o i n t e g r a t e t h e r a t e e q u a t i o n s .

As a r e s u l t of a r e c e n t i n t e r e s t i n heavy-ion driven ICF t a r g e t s , we a r e improving t h e stopping power c a l c u l a t i o n s used i n t h e simulation codes f o r t a r g e t design. Knowing t h e e f f e c t i v e charge of t h e p r o j e c t i l e ion i s very important f o r t h i s s i n c e t h e stopping power i s p r o p o r t i o n a l t o t h e square of t h e e f f e c t i v e charge. Because of t h e extreme c o n d i t i o n s of heavy-ion i r r a d i a t i o n ( a very heavy p r o j e c t i l e a t very high v e l o c i t y e n t e r s t h e t a r g e t i n a low charge s t a t e ) , we a r e concerned t h a t t h e e f f e c t i v e charge may s i g n i f i c a n t l y d i f f e r from t h e usual e q u i l i b r i u m charge s t a t e . This a r t i c l e d i s c u s s e s an approximate c a l c u l a t i o n of t h e charge s t a t e of a f a s t , heavy-ion t r a v e r s i n g a f i n i t e temperature t a r g e t plasma.

*work performed under t h e a u s p i c e s of t h e U.S. Department o f Energy by t h e Lawrence Livermore N a t i o n a l L a b o r a t o r y under c o n t r a c t W-7405-ENG-48.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983810

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To p e r f o r m t h i s c a l c u l a t i o n , we u s e an average atom model f o r t h e p r o j e c t i l e i o n t o g e t h e r w i t h s c a l e d h y d r o g e n i c r a t e s 2 f o r t h e c o n t i n u ~ m 1 and bound processes ( l i s t e d i n Table I ) t o f o l l o w t h e t i m e h i s t o r y o f t h e p r o j e c t i l e charge s t a t e . S i n c e o u r m a i n i n t e r e s t l i e s i n t h e case o f a high-Z p r o j e c t i l e and a warm low-Z t a r g e t , we have e x c l u d e d v a r i o u s resonances processes t h a t a r e i m p o r t a n t f o r near n e u t r a l i o n s and enhanced charge t r a n s f e r f r o m t a r g e t t o p r o j e c t i l e where t h e r e i s a c o i n c i d e n c e i n t h e e n e r g y l e v e l s i n v o l v e d . I n o u r case, s i n c e we e x p e c t t h e t a r g e t and p r o j e c t i l e atoms t o be h i g h l y i o n i z e d , such resonance e f f e c t s a r e s m a l l .

TABLE I

ELECTRON RATES

o BOUND-BOUND E X C l T A T l O N & UE-EXClTAT 1ON

-

o BOUND-FREE I O N I Z A T I O N

-

0 R A D l A T l V E RECOMBINATION

-

0 1 3 - B O D Y RECOMB I N A T I O N )

1 0 8

- ⁰ BOUND-BOUND E X C I T A T I O N & D € - E X C I T A T l O N RATES - o BOUND-FREE I O N I Z A T I O N

Next we p r e s e n t arguments t o show t h a t even a t % s o l i d d e n s i t y i n t h e t a r g e t , t h r e e body r e c o m b i n a t i o n i s s t r o n g l y suppressed b y k i n e m a t i c e f f e c t s . To c a l c u l a t e t h e r a t e f o r t h i s process, we must i n t e g r a t e t h e c r o s s - s e c t i o n o v e r t h e i n i t i a l e l e c t r o n d i s t r i b u t i o n f u n c t i o n i n e n e r g y and angle. I n t h e r e f e r e n c e frame o f t h e p r o j e c t i l e , t h e s e d i s t r i b u t i o n s a r e e s s e n t i a l l y d e l t a - f u n c t i o n s . T o g e t h e r w i t h r e q u i r e m e n t s o f e n e r g y and momenturn

c o n s e r v a t i o n and t h e f o r m of t h e c r o s s - s e c t i o n , t h i s l e a d s t o a s m a l l r e s u l t . To g e t a q u a n t i t a t i v e e s t i m a t e , we proceed as f o l l o w s . From d e t a i l e d balance, we c a n r e l a t e t h e r e c o m b i n a t i o n d i f f e r e n t i a l c r o s s - s e c t i o n t o t h a t f o r i o n i z a t i o n . U s i n g a Born a p p r o x i m a t i o n f o r t h e i o n i z a t i o n d i f f e r e n t i a l c r o s s - s e c t i o n , and p e r f o r m i n g t h e phase space i n t e g r a l , we f i n a l l y o b t a i n an a n a l y t i c e x p r e s s i o n f o r t h e t h r e e - b o d y r a t e . There a r e two p a r t s t o t h i s : an

a n g u l a r p a r t shown i n F i g . 1 and an e n e r g y f a c t o r l i s t e d i n s c a l e d f o r m i n Table 11. Even a t t h r e s h o l d ( € / I z 1, a p p r o p r i a t e a t e q u i l i b r i u m ; where E i s t h e e q u i v a l e n t e l e c t r o n energy, and I is t h e i o n i z a t i o n e n e r g y o f t h e l e a s t bound e l e c t r o n ) , t h e c o m b i n a t i o n o f a n g u l a r and e n e r g y f a c t o r s y i e l d s a s u p p r e s s i o n o f 1000-10,000 o f t h r e e body r e l a t i v e t o r a d i a t i v e r e c o m b i n a t i o n f o r f a s t i o n s o f h i g h charge.

Ion energy (MeV/amu)

F i g u r e 1. P l o t o f t h e dependence o f t h r e e - b o d y r e c o m b i n a t i o n r a t e on t h e a n g u l a r p o i n t o f t h e phase space as a f u n c t i o n o f p r o j e c t i l e i o n energy f o r a f r e e e l e c t r o n t e m p e r a t u r e o f 100 eV, 300 eV, and 1 keV.

F i n a l l y , we remark t h a t we i n c l u d e t h e i o n - i o n processes b y z - s c a l i n g f r o m t h e e l e c t r o n r e s u l t s . As shown i n F i g . 2, t h i s i s a good a p p r o x i m a t i o n f o r f a s t i o n s on f u l l y s t r i p p e d t a r g e t s w h i c h i s t h e case o f g r e a t e s t i n t e r e s t here.

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TABLE I 1

COMPARISON O F THE THREE-BODY RECOMB INATION RATE TO THE RADIATIVE RECOMBINATION RATE

TNREE - BODY RECOMBINATION R A T E Ne N 4

= R0 E l 1 - -

RADlATlVE RECOMBINATION RATE No Z*

C

.-

U

S

g

^-

L

0 -

S

.-

m -

,- N f

0

+

o e + Au ⁺ 2e + Au (E.J. McGuire) -

=. P + ^{AU + P} + ^e

+

I 0-77 I I l ! I I ! l

10-2 0.1 1

Electron energy (keV)

F i g u r e 2. A c o m p a r i s o n o f total ionization cross-sections f o r incident electrons and protons as a f u n c t i o n of electron e n e r g y (or ( E /M ) ^X M e f o r protons).

P P

I n summary, t h e i m p o r t a n t processes are c o l l i s i o n a l i o n i z a t i o n balanced a t e q u i l i b r i u m by r a d i a t i v e recombination. Although t h i s sounds s i m i l a r t o corona1 e q u i l i b r i u m , i n t h e present case t h e narrow e l e c t r o n energy

d i s t r i b u t i o n means t h a t t h e same e l e c t r o n s e x c i t e as recombine, whereas w i t h Maxwellian e l e c t r o n s , i t i s t h e h i g h energy ones t h a t do t h e e x c i t a t i o n and t h e low energy ones t h a t recombine.

The r e s u l t o f s o l v i n g t h e r a t e equations f o r v a r i o u s p r o j e c t i l e i o n s w i t h an energy o f 46 MeV/Amu (corresponding t o 9 Gev f o r a g o l d i o n ) i s shown i n F i g . 3a f o r e l e c t r o n - i o n e x c i t a t i o n and i n F i g . 3b f o r i o n - i o n e x c i t a t i o n f o r a t a r g e t i o n w i t h a Z* o f 10. For such f a s t ions, t h e e q u i l i b r i u m charge s t a t e i s very c l o s e t o f u l l y s t r i p p e d i n a l l cases, a l t h o u g h t h e t i m e t o reach e q u i l i b r i u m increases w i t h p r o j e c t i l e Z. Furthermore, a l t h o u g h t h e

e q u i l i b r i u m t i m e decreases f o r t h e i o n - i o n case ( F i g . 3b), t h e e q u i l i b r i u m charge i s almost i d e n t i c a l t o t h e e l e c t r o n - i o n case, i n agreement w i t h t h e Bohr c r i t e r i o n . Also note t h a t t h e t i m e t o s t r i p t h e f i r s t e l e c t r o n s i s very s h o r t , and hence decreasing t h e i n i t i a l s t r i p p i n g t i m e i s n o t important. To express t h e t i m e e q u i l i b r a t i o n t i m e i n more p h y s i c a l u n i t s , we l i s t i n Table 111 t h e e q u i l i b r a t i o n d i s t a n c e ( f o r a c o n s t a n t v e l o c i t y i o n ) i n u n i t s o f t h e c o l d range f o r t h e ions i n Fig. 3. The d e l a y d i s t a n c e increases r a p i d l y w i t h p r o j e c t i l e Z, and f o r g o l d ions i s 1/6 o f t h e t o t a l range, a s u b s t a n t i a l e f f e c t .

TABLE 111

QUILIBRATION DISTANCE ( I N UNITS OF COLD RANGE)

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0

10-5 10-4 10-3 10-2 10-1 1 10 Time (picoseconds) a)

I 1 1 l " ] l " ] ' " 1 ^{l l ' l 1 l *}

- -

- Au

- N e = 6 X 1 0 ~ ~ c m - ~

- E,,, = 46 MeV/Amu

- Electron

-

- - - - - - -

-

- -

E,,, = 46 MeVIAmu Ion

-

-

10-5 10-4 10-3 10-2 10-I 1 10 Time fpico seconds) b 1

F i g u r e 3. a,b Time dependent charge s t a t e f o r s e v e r a l i o n s f o r e l e c t r o n c o l l i s i o n s ( F i g . 3a) and i o n c o l l i s i o n s ( F i g . 3b) f o r a t a r g e t c o r r e s p o n d i n g t o f u l l y s t r i p p e d aluminum a t s o l i d d e n s i t y .

The c a l c u l a t i o n s r e p o r t e d h e r e r e f e r a f r e e e l e c t r o n d e n s i t y 2. 6 X

1 0 Cm-3, c o r r e s p o n d i n g t o f u l l y i o n i z e d aluminum a t n e a r s o l i d d e n s i t y .

The s t r i p p i n g r a t e i s p r o p o r t i o n a l t o t h e e l e c t r o n d e n s i t y and t h e e n e r ~ y - l ~ s s r a t e i s a p p r o x i m a t e l y l i n e a r i n t h e e l e c t r o n d e n s i t y .

T h e r e f o r e , we b e l i e v e t h e s e r e s u l t s a r e a l s o v a l i d f o r high-Z t a r g e t s . N o t e t h a t t h e o m i t t e d e f f e c t s ( c h a r g e t r a n s f e r , e t c . ) i n c r e a s e r e c o m b i n a t i o n , and hence l e n g t h e n t h e e q u i l i b r i u m d i s t a n c e . The m a i n e f f e c t t h a t would s h o r t e n t h e s t r i p p i n g d i s t a n c e i s t h e i o n - i o n c o l l i s i o n s ; hence as t h e t a r g e t warms up, t h e e f f e c t w i l l decrease.

By r e p e a t i n g t h e c a l c u l a t i o n s o f F i g . 3 f o r l o w e r e n e r g y ions, we can o b t a i n t h e e q u i l i b r i d m charge f o r each i o n as a f u n c t i o n o f p r o j e c t i l e

v e l o c i t y . These r e s u l t s a r e shown i n F i g . 4a and 4b f o r t h e same i o n s used i n F i g . 3. We a l s o show f o r comparison t h e s e m i - e m p i r i c a l B e t z f o r m ~ l a . ~ A t h i g h v e l o c i t i e s , t h e agreement i s v e r y good; a t medium v e l o c i t i e s s h e l l e f f e c t s cause o u r r e s u l t s t o d e v i a t e f r o m Betz, and a t l o w v e l o c i t i e s (where t h e B e t z f o r m u l a s t a r t s t o f a i l ) o u r r e s u l t s a r e s y s t e n i a t i c a l l y h i g h .

F i n a l l y , we m e n t i o n a u s e f u l r e s u l t t h a t we o b t a i n as a b y - p r o d u c t of s o l v i n g t h e r a t e e q u a t i o n s , namely t h e r a d i a t i o n e m i s s i o n o f t h e p r o j e c t i l e

i o n as i t t r a v e r s e s t h e t a r g e t . We show i n F i g . 5 t h e t i m e i n t e g r a t e d e m i s s i o n spectrum o f a c o n s t a n t v e l o c i t y 46 MeVIHmu g o l d i o n p e n e t r a t i n g somewhat more t h a n a c o l d t a r g e t range depth. S i n c e t h e t o t a l r a d i a t i o n e m i s s i o n i s % 2% o f t h e o r i g i n a l i o n energy, i t i s a p o t e n t i a l source o f p r e h e a t i n t h e r e s t o f t h e t a r g e t .

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Ion v e l o c i t y ( v l c ) a)

I o n v e l o c i t y ( v l c ) b 1

F i g u r e 4. a,b E q u i l i b r i u m charge s t a t e as a f u n c t i o n o f i o n v e l o c i t y f o r t h e low Z ( F i g . 4a) and h i g h Z ( F i g . 4b) i o n s o f F i g . 3.

1 10 100 Photon energy (IceV)

Figure 5. Time integrated emission spectrum f o r a gold ion traversing 0.25 g/cm of cold Al. 2

DISCLAIMER

This document was prepared as an account of work sponsored by an agency of the United S t a t e s Government. Neither the United S t a t e s Government nor the University of California nor any of t h e i r employees, makes any warranty, express or implied, or assumes any legal l i a b i l i t y or r e s p o n s i b i l i t y f o r t h e accuracy, completeness, o r usefulness of any information, apparatus, product, or process disclosed, or represents t h a t i t s use would not infringe p r i v a t e l y owned rights. Reference herein t o any s p e c i f i c commercial products, process, o r service by t r a d e name, trademark,, manufacturer, or otherwise, does not necessarily c o n s t i t u t e o r imply i t s endorsement, recommendation, o r favoring by f? United S t a t e s Government o r t h e University of California. The views and opinions of authors expressed herein do not necessarily s t a t e o r r e f l e c t those of the United S t a t e s Government thereof, and s h a l l not be used f o r advertising o r product endorsement purposes.

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REFERENCES

1. W. A. Lokke and W. H. Grasberger, UCRL 52276 (1977); R. More, UCRL 84991 (1981); G. Zimmerman and R. More, JQSRT 23, 517 (1980), R. More, JQSRT 27, 345 (1982).

-

2. D. H. Sampson and L. B. Goldon, Ap. J.

161,

3. H. D. Betz, Rev. Mod. Phys. 44, 465 (1972).